Class 10 Mathematics Chapter 9 Matrices

Key Concept: Identity Matrix, Non-Commutative Matrices

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion states that $(A + B)(A - B) = A^2 - B^2$ only when $AB = BA$. Let's verify this:
$(A + B)(A - B) = A^2 - AB + BA - B^2$
For this to equal $A^2 - B^2$, $-AB + BA$ must be zero, which implies $AB = BA$. Thus, the Assertion is true.
The Reason states that matrix multiplication is commutative for identity matrices. Since $I \times A = A \times I$ for any matrix $A$ of the same order as $I$, the Reason is also true.
However, the Reason does not explain the Assertion because the Assertion is a general condition for all matrices where commutativity holds, while the Reason specifically refers to identity matrices. Hence, both are true, but the Reason is NOT the correct explanation of the Assertion.

Key Concept: Identity Matrix for Multiplication

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that multiplying any $2 \times 2$ matrix $A$ by the $2 \times 2$ identity matrix $I$ results in the same matrix $A$. This is true as per the definition of the identity matrix. The reason explains why this happens: the identity matrix leaves any matrix unchanged when multiplied by it, which directly explains the assertion. Both statements are true, and the reason correctly explains the assertion.

Key Concept: Identity Matrix for Multiplication

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that multiplying any square matrix $A$ by the identity matrix $I_n$ results in the same matrix $A$, which is a fundamental property of identity matrices. The reason correctly describes the structure of the identity matrix, which indeed has 1s on the main diagonal and 0s elsewhere. This specific structure is precisely why the identity matrix leaves other matrices unchanged when multiplied. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Non-Commutativity of Matrix Multiplication, Zero Matrix

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion claims that if the product of two matrices $A$ and $B$ is the zero matrix, then at least one of them must be the zero matrix. However, this is not always true. There exist non-zero matrices whose product is zero, as demonstrated in the syllabus example where $A = \begin{bmatrix} -3 & 3 \\ 4 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & 6 \\ -2 & -3 \end{bmatrix}$ yield $AB = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$. Thus, the assertion is false. The reason correctly states that the product of two non-zero matrices can indeed be a zero matrix, which is true based on the given example. Therefore, the assertion is false, but the reason is true.

Key Concept: Matrix Multiplication Properties

d) Assertion is false, but Reason is true.
[Solution Description]
Let us verify the assertion using an example. Consider $A = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$. Multiplying them:
$AB_{11} = (1 \times 1) + (-1 \times 1) = 0$
$AB_{12} = (1 \times 1) + (-1 \times 1) = 0$
$AB_{21} = (-1 \times 1) + (1 \times 1) = 0$
$AB_{22} = (-1 \times 1) + (1 \times 1) = 0$
Thus, $AB = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$, even though neither $A$ nor $B$ is a zero matrix. This shows that the Assertion is false. However, the Reason correctly states that the product of two non-zero matrices can indeed be a zero matrix, which is true.

Key Concept: Matrix Multiplication

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Matrix $A$ is an identity matrix of order $2 \times 2$. According to the property of the identity matrix, multiplying any matrix by the identity matrix results in the original matrix. Therefore, the assertion (A) is true. The reason (R) correctly explains why the assertion is true as it states a fundamental property of identity matrices.

Key Concept: Scalar Multiplication, Associative Property

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's take an example. Let $A = \begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix}$, $k = 4$, and $m = 5$.
Step 1: Calculate $mA = 5 \times \begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} 10 & -5 \\ 0 & 15 \end{bmatrix}$.
Step 2: Now multiply by $k$: $k(mA) = 4 \times \begin{bmatrix} 10 & -5 \\ 0 & 15 \end{bmatrix} = \begin{bmatrix} 40 & -20 \\ 0 & 60 \end{bmatrix}$.
Step 3: Calculate $(km)A = (4 \times 5) \times A = 20 \times \begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} 40 & -20 \\ 0 & 60 \end{bmatrix}$.
The results from Step 2 and Step 3 are identical, proving that $k(mA) = (km)A$. Therefore, the Assertion is true.
The Reason states that scalar multiplication is associative with respect to multiplication of scalars, which is a fundamental property mentioned in the syllabus. Hence, the Reason is also true and correctly explains the Assertion.

Key Concept: Multiplication of a Matrix by a Scalar

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that multiplying matrix $A = \begin{bmatrix} 5 & -2 \\ 3 & 1 \end{bmatrix}$ by scalar $-1$ results in $\begin{bmatrix} -5 & 2 \\ -3 & -1 \end{bmatrix}$. To verify this, multiply each element of $A$ by $-1$:
1. Multiply the first row first column: $5 \times (-1) = -5$
2. Multiply the first row second column: $-2 \times (-1) = 2$
3. Multiply the second row first column: $3 \times (-1) = -3$
4. Multiply the second row second column: $1 \times (-1) = -1$
Thus, the result is $\begin{bmatrix} -5 & 2 \\ -3 & -1 \end{bmatrix}$, which matches the assertion. The reason explains the process correctly: each element is indeed multiplied by the scalar. Hence, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Multiplication of a Matrix by a Scalar

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this, we multiply each element of the given matrix by the scalar 5. The first element 2 becomes $5 \times 2 = 10$, the second element $-1$ becomes $5 \times (-1) = -5$, the third element 0 remains $5 \times 0 = 0$, and the fourth element 3 becomes $5 \times 3 = 15$. Hence, the resulting matrix is $\begin{bmatrix} 10 & -5 \\ 0 & 15 \end{bmatrix}$. The reason correctly explains that each element is multiplied individually by the scalar.

Key Concept: SOLVING MATRIX EQUATIONS

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve the assertion and reason, we analyze both statements separately.
1. **Assertion (A):** The statement claims that the solution to $A + X = B$ is $X = B - A$. This is true because subtracting $A$ from both sides of the equation gives $X = B - A$, provided all matrices are of the same order.
2. **Reason (R):** The reason states that matrix addition and subtraction are defined only for matrices of the same order. This is also correct because the operations require corresponding elements to add or subtract, which is possible only if dimensions match.
The Reason correctly explains the condition under which the Assertion holds true, as the solution $X = B - A$ is valid only when $A$ and $B$ have the same order.

Key Concept: Matrix Equations

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that for matrices $A$, $B$, and $X$ of the same order, if $A + X = B$, then $X = B - A$. This is correct based on the properties of matrix operations. The reason states that matrix subtraction is only defined when the matrices involved are of the same order. This is also true because matrix operations require compatible dimensions. However, the reason does not directly explain why $X = B - A$; it merely states a condition necessary for the subtraction to be valid. Therefore, while both statements are true, the reason is not the correct explanation of the assertion.

Key Concept: Matrix Equation Solving

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for the equation $A + X = B$, the solution is obtained by subtracting matrix $A$ from matrix $B$ to find $X$. This is correct as per the given syllabus. The reason explains the procedure, which is that matrix subtraction is performed element-wise and requires the matrices to be of the same order, which is true. Moreover, the reason correctly explains why the assertion holds since it describes the underlying operation needed to solve the matrix equation.

Key Concept: Additive Inverse, Matrix Properties

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, analyze the assertion: Given $A + B = O$, we can rearrange it to $B = -A$, proving that $B$ must indeed be $-A$ for the equation to hold. Therefore, the assertion is true.
Next, consider the reason: The additive inverse of $A$ is defined as $-A$ because $A + (-A) = O$ holds uniquely, meaning no other matrix except $-A$ satisfies this condition. Thus, the reason is true and correctly explains the assertion.

Key Concept: Additive Inverse of Matrices

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states that the additive inverse of matrix $A$ is $-A$, which is true as per the definition of additive inverse in matrices. However, the reason claims that adding a matrix to its additive inverse yields the identity matrix, which is false. The correct result should be the null matrix, not the identity matrix. Therefore, the assertion is true, but the reason is false.

Key Concept: Additive Inverse

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the additive inverse of a matrix $A$ is $-A$. According to the syllabus, this is correct as the additive inverse of a matrix is indeed its negative. The reason states that adding $A$ and $-A$ results in the null matrix $O$, which is also correct as per the syllabus. Furthermore, the reason correctly explains why $-A$ is the additive inverse of $A$ because their sum yields the null matrix. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Additive Identity in Matrices

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that adding a null matrix $O$ to any $2 \times 3$ matrix $A$ results in the same matrix $A$. This is true because the null matrix does not change the original matrix when added to it. The reason given is that the null matrix $O$ acts as the additive identity for matrices of the same order, which is also true. Moreover, the reason correctly explains why the assertion holds since the property of the null matrix being an additive identity directly implies $A + O = A$. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Additive Identity and Additive Inverse

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that for any matrix $A$, there exists an additive inverse $-A$ such that their sum is the null matrix $O$. This is true because by definition, the additive inverse of a matrix $A$ is $-A$ (obtained by negating each element of $A$) and $A + (-A) = O$.
The reason states that the null matrix $O$ is the additive identity for matrices. This is also true because adding $O$ to any matrix $A$ leaves $A$ unchanged ($A + O = A$). However, the reason does not explain why the additive inverse exists; it simply states a property of the null matrix. Thus, both are true, but the reason does not explain the assertion.

Key Concept: Additive Identity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for any matrix $A$, adding the null matrix $O$ to it results in the same matrix $A$. This is true because the null matrix has all elements as zero, and adding zero does not change the original matrix. The reason correctly explains that the null matrix $O$ serves as the additive identity, as it leaves the original matrix unchanged when added to it. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Equality of Matrices, Solving Matrix Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if two matrices are equal, their corresponding elements generate solvable equations. This is true because matrix equality requires identical dimensions and corresponding elements. For example, if $\begin{bmatrix} x + y & 2 \\ 3 & x - y \end{bmatrix} = \begin{bmatrix} 4 & 2 \\ 3 & 0 \end{bmatrix}$, then solving $x + y = 4$ and $x - y = 0$ gives $x = 2$, $y = 2$.
The reason correctly explains that matrix equality ensures identical elements, which directly leads to solvable equations. Thus, the reason is the correct explanation for the assertion.

Key Concept: Equality of Matrices

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion (A) states that for two matrices to be equal, their corresponding elements must be equal, which is true as per the definition of matrix equality. The reason (R) states that two matrices can only be equal if they have the same order, which is also true because matrices of different orders cannot be compared for equality. However, the reason does not explain why corresponding elements must be equal; it merely states a necessary condition for equality. Therefore, both the assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Equality of Matrices

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if two matrices are equal, they must have the same order. This is true because one of the conditions for matrix equality is having the same order. The reason states that equality of matrices requires both the same order and identical corresponding elements. This is also true and correctly explains why the assertion is valid since the condition mentioned in the assertion (same order) is part of the broader requirement stated in the reason.

Key Concept: Equality and Transpose of Matrices

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion and reason:
1. **Assertion**: If two matrices $A$ and $B$ are equal, they must have the same order and corresponding elements. Taking the transpose involves interchanging rows and columns, but it does not change the values of the elements or the overall equality condition. Hence, $A = B$ implies $A' = B'$. Thus, the Assertion is true.
2. **Reason**: The transpose operation indeed preserves the equality of matrices because it only rearranges the elements by swapping rows and columns, not changing their values. Therefore, the Reason correctly explains why the Assertion holds true.
Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Equality of Matrices

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
For two matrices to be equal, they must satisfy two conditions: (1) They must have the same order, and (2) Their corresponding elements must be identical. The Assertion states that if two matrices are equal, they must have the same order, which is true because equality requires matching dimensions. The Reason states that two matrices are equal if and only if their corresponding elements are identical, which is also true but incomplete without mentioning the condition of having the same order. Therefore, both Assertion and Reason are true, but Reason alone does not fully explain Assertion.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the matrix $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ has exactly 4 elements. This is true because the matrix has 2 rows and 2 columns, so the total number of elements is $2 \times 2 = 4$.
The reason states that the number of elements in a matrix is the product of its number of rows and columns, which is also true. Furthermore, this correctly explains why the assertion is true, as it provides the mathematical basis for counting the elements in the given matrix.
Therefore, both the Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Applications of Matrices

d) Assertion is false, but Reason is true.
[Solution Description]
To determine the correctness of the assertion and reason, we analyze them step-by-step:
1. **Assertion Analysis**: The assertion states that the matrix representation of a system of linear equations is always solvable using the inverse matrix method if the coefficient matrix is square. This is not entirely true because even if the coefficient matrix is square, the system may not be solvable if the determinant of the matrix is zero (i.e., the matrix is singular). Hence, the assertion is false.
2. **Reason Analysis**: The reason states that a square matrix with a non-zero determinant has an inverse, which can be used to solve the associated system of linear equations. This is a true statement as per matrix theory.
3. **Conclusion**: Since the assertion is false but the reason is true, the correct option is (d).

Key Concept: Equality of Matrices

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
For two matrices to be equal, they must satisfy two conditions: (1) They must have the same order, i.e., the same number of rows and columns. (2) Their corresponding elements must be equal. Therefore, the Assertion (A) is true because it states that corresponding elements must be equal for matrix equality. The Reason (R) is also true as it correctly states that equality of matrices requires them to have the same order. However, the Reason does not explain why corresponding elements must be equal (Assertion). Thus, both are true, but the Reason is not the correct explanation of the Assertion.

Key Concept: Solving Systems of Linear Equations

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that matrices can solve systems of linear equations efficiently, which is true as per the syllabus. The reason explains that representing the system in matrix form simplifies computations, which is also true. Moreover, the reason correctly justifies the assertion since matrix representation is indeed the basis for efficient solving methods like Gaussian elimination. Thus, both are true, and the reason is the correct explanation.

Key Concept: Properties of Determinants, Matrix Multiplication

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion involves properties of determinants and matrix multiplication.
Step 1: Given $\det(A) = 4$ and $\det(B) = -2$.
Step 2: The determinant of the transpose of a matrix is equal to the determinant of the original matrix, so $\det(B^T) = \det(B) = -2$.
Step 3: The determinant of the product of two matrices is the product of their determinants, so $\det(AB^T) = \det(A) \cdot \det(B^T) = 4 \times (-2) = -8$.
Step 4: The reason correctly explains the assertion because it states the property used in Step 3.
Thus, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Determinant of Transpose and Singular Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To solve this assertion-reason question, let us analyze both statements step by step.
1. **Assertion (A)**: "If the determinant of a matrix $A$ is zero, then the determinant of its transpose $A^T$ is also zero."
- From the property of determinants, we know that $|A| = |A^T|$. So, if $|A| = 0$, then $|A^T| = 0$ must also hold. Therefore, Assertion (A) is true.
2. **Reason (R)**: "The determinant of a matrix is equal to the determinant of its transpose."
- This is a fundamental property of determinants, as stated in the syllabus: $|A| = |A^T|$. Thus, Reason (R) is true.
3. **Explanation Check**: Since the truth of Assertion (A) is directly based on the property given in Reason (R), Reason (R) correctly explains Assertion (A).
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Properties of Determinants

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The given matrix $A$ is a diagonal matrix with diagonal elements $5$ and $5$. According to the property of determinants, the determinant of a diagonal matrix is the product of its diagonal elements. Here, $5 \times 5 = 25$. Thus, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Determinants, Singular Matrices

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if the determinant of a square matrix $A$ is zero, then $A$ is singular. This is true because a matrix is singular if and only if its determinant is zero. The reason states that a singular matrix does not have an inverse, which is also true. Furthermore, the reason correctly explains why the assertion is true: a matrix with a zero determinant cannot be inverted, hence it is singular. Thus, both assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Determinant Properties

c) Assertion is true, but Reason is false.
[Solution Description]
The assertion states a fundamental property of determinants: $\det(AB) = \det(A)\det(B)$ for square matrices. This is true. The reason claims it is due to elementary row operations preserving the determinant, which is incorrect—the property holds due to multiplicative behavior of determinants, not solely row operations.
Step-by-step reasoning:
1) For any two $n \times n$ matrices $A$ and $B$, $\det(AB) = \det(A)\det(B)$. (Theorem in linear algebra.)
2) Elementary row operations affect the determinant (e.g., row swapping multiplies by $-1$, scaling multiplies by a factor), but preservation is not the basis for the product rule.
3) Thus, the assertion is true, but the reason is false.

Key Concept: Determinant of 2x2 Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description] To verify the assertion, calculate the determinant of $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ using the formula for a $2 \times 2$ matrix: $\text{det}(A) = (1)(4) - (2)(3) = 4 - 6 = -2$. The assertion is true. The reason states the correct formula for the determinant of a $2 \times 2$ matrix and is also true. Additionally, the reason correctly explains why the assertion holds true.

Key Concept: Additive Inverse, Uniqueness of Additive Inverse

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the additive inverse of a matrix $A$ is unique and is given by $-A$. This is true because according to the definition of additive inverse, for any matrix $A$, there exists exactly one matrix $B$ (which is $-A$) such that $A + B = B + A = O$. The reason correctly explains this uniqueness property by stating that only one matrix $B$ satisfies the condition $A + B = B + A = O$. Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Definition of Inverse

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for any $2 \times 2$ matrix $A$, its additive inverse is $-A$. This is correct because the definition of additive inverse requires $A + (-A) = O$, where $O$ is the null matrix.
The reason states that adding a matrix to its additive inverse results in the null matrix. This is also true and directly explains why the assertion is true, as it aligns with the definition of additive inverses in matrices.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Definition of Additive Inverse

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
According to the definition of additive inverse, for any matrix $A$, the additive inverse is $-A$. When $A$ is added to $-A$, the result must be the null matrix. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Additive Inverse, Matrix Equations

c) Assertion is true, but Reason is false.
[Solution Description]
The Assertion states that for any square matrix $A$, the equation $A + (-A) = 0$ holds true, which defines $-A$ as the additive inverse of $A$. This is correct as per the properties of additive inverses in matrices. The Reason claims that the additive inverse is unique due to the commutativity of matrix addition. While it is true that the additive inverse is unique and matrix addition is commutative, commutativity does not directly explain the uniqueness of the additive inverse. Uniqueness follows from the definition of additive inverses and the properties of matrix operations, not just commutativity. Therefore, the Assertion is true, but the Reason is false.

Key Concept: Inverse of a Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that if $A$ is the additive inverse of $B$, then $B$ must be the additive inverse of $A$. This is true because, by definition, the additive inverse of a matrix $A$ is $-A$, and $-(-A) = A$, meaning $A$ and $-A$ are inverses of each other. The reason states that the sum of a matrix and its additive inverse is the zero matrix, which is also true by definition. Moreover, the reason correctly explains the assertion since the property of additive inverses implies symmetry in their relationship.

Key Concept: Inverse of a Matrix

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that a square matrix with determinant zero has an inverse, which is false because the condition for a matrix to have an inverse is that its determinant must be non-zero. The Reason correctly states this condition. Therefore, the Assertion is false, but the Reason is true.

Key Concept: Skew-Symmetric Matrix, Eigenvalues

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
A skew-symmetric matrix satisfies $A^T = -A$. For any eigenvalue $\lambda$ of $A$ with eigenvector $v$, we have $Av = \lambda v$. Taking the conjugate transpose, $v^* A^T = \overline{\lambda} v^*$. Multiplying by $v$ gives $v^* A^T v = \overline{\lambda} v^* v$. Substituting $A^T = -A$, $-v^* A v = \overline{\lambda} v^* v$. Since $Av = \lambda v$, this becomes $-v^* (\lambda v) = \overline{\lambda} v^* v \Rightarrow -\lambda \|v\|^2 = \overline{\lambda} \|v\|^2 \Rightarrow \lambda = -\overline{\lambda}$. Thus, $\lambda$ must be purely imaginary. The reason correctly states the property of skew-symmetric matrices but does not explain why eigenvalues are purely imaginary.

Key Concept: Matrix Properties

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, let's verify the assertion: By definition, a square matrix $A$ is skew-symmetric if its transpose equals its negative ($A^T = -A$). This part is correct.
Now examine the reason: For any diagonal element $a_{ii}$ in $A$, when we take the transpose, $a_{ii}^T = a_{ii}$. From the skew-symmetric condition, this equals $-a_{ii}$. So $a_{ii} = -a_{ii}$ which implies $2a_{ii} = 0$ or $a_{ii} = 0$. Therefore all diagonal elements must be zero.
The reason correctly explains why diagonal elements are zero in a skew-symmetric matrix, based on the assertion's definition.

Key Concept: Skew-Symmetric Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states the definition of a skew-symmetric matrix ($A = -A^T$), which is correct. The reason is also true because taking the transpose of a skew-symmetric matrix flips the sign of off-diagonal elements but leaves diagonal elements unchanged. Hence, $a_{ii} = -a_{ii}$ implies $a_{ii} = 0$. The reason correctly explains why diagonal elements must be zero for the assertion to hold.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that any square matrix can be uniquely decomposed into the sum of a symmetric and a skew-symmetric matrix. This is true because for any square matrix $A$, we can define:
Symmetric part: $P = \frac{A + A^T}{2}$. Transpose of $P$ is $P^T = \frac{A^T + A}{2} = P$, so $P$ is symmetric.
Skew-symmetric part: $Q = \frac{A - A^T}{2}$. Transpose of $Q$ is $Q^T = \frac{A^T - A}{2} = -\left(\frac{A - A^T}{2}\right) = -Q$, so $Q$ is skew-symmetric.
Summing these parts gives $A = P + Q$. The uniqueness follows from the fact that if $A$ were written as another sum $A = P' + Q'$, then $P - P' = Q' - Q$, where the left side is symmetric and the right side is skew-symmetric. The only matrix both symmetric and skew-symmetric is the zero matrix, so $P = P'$ and $Q = Q'$.
The reason correctly provides the explicit forms of the symmetric and skew-symmetric parts of $A$ and also correctly explains why they are symmetric and skew-symmetric respectively. Therefore, both assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Symmetric and Skew-Symmetric Matrices

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, verify the assertion (A):
Given matrix $A = \begin{bmatrix} 0 & -3 \\ 3 & 0 \end{bmatrix}$.
Transpose of $A$ is $A^T = \begin{bmatrix} 0 & 3 \\ -3 & 0 \end{bmatrix}$.
Now, $-A^T = -\begin{bmatrix} 0 & 3 \\ -3 & 0 \end{bmatrix} = \begin{bmatrix} 0 & -3 \\ 3 & 0 \end{bmatrix}$, which equals $A$. Hence, $A$ is skew-symmetric.
Next, verify the reason (R):
Property of a skew-symmetric matrix states that its diagonal elements must be zero ($a_{ii} = 0$). Here, the diagonal elements of $A$ are $0$ and $0$, satisfying this property.
The reason correctly explains why $A$ is skew-symmetric because it satisfies the necessary condition for a skew-symmetric matrix.

Key Concept: Symmetric Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To determine whether the given matrix $A$ is symmetric, we first find its transpose:
$A^T = \begin{bmatrix} 1 & 5 \\ 5 & -3 \end{bmatrix}$
Comparing $A$ and $A^T$, we observe that they are equal ($A = A^T$). Therefore, the Assertion is true.
The Reason states the definition of a symmetric matrix: a square matrix is symmetric if it equals its transpose. Since this correctly explains why $A$ is symmetric, the Reason is also true and correctly explains the Assertion.

Key Concept: Transpose of a Matrix, Properties of Transpose

c) Assertion is true, but Reason is false.
[Solution Description]
(1) According to the properties of transpose, $(A^T)^T = A$ is true for any matrix (square or non-square). So Assertion (A) is true.
(2) The reason claims that the transpose operation reverses the order of elements along the principal diagonal. However, this is false because the elements on the principal diagonal remain unchanged in a transpose operation (as they lie on the same row and column index).
(3) Therefore, Assertion is true but Reason is false.

Key Concept: Transpose of a Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let's analyze it step by step:
1. Let $A$ be any square matrix and $A'$ its transpose.
2. Consider the matrix $A + A'$. To check if it is symmetric, we need to verify whether $(A + A')' = A + A'$.
3. Using the property of transpose: $(A + A')' = A' + (A')' = A' + A$ (since the transpose of a transpose gives the original matrix).
4. Since matrix addition is commutative ($A' + A = A + A'$), we have $(A + A')' = A + A'$.
5. Thus, $A + A'$ is symmetric.
The reason states the property $(A + B)' = A' + B'$, which is indeed true and correctly explains why the assertion holds. Therefore, both assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Transpose of a Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the transpose of a $3 \times 2$ matrix is a $2 \times 3$ matrix, which is correct because transposing swaps rows and columns. The reason explains why this happens by stating that the order changes when rows and columns are interchanged. Since both statements are true and the reason correctly explains the assertion, the correct answer is option (a).

Key Concept: Matrix Multiplication, Non-Commutativity

d) Assertion is false, but Reason is true.
[Solution Description]
First, let's compute $AB$:
$AB = \begin{bmatrix} 1 \cdot 5 + 2 \cdot 7 & 1 \cdot 6 + 2 \cdot 8 \\ 3 \cdot 5 + 4 \cdot 7 & 3 \cdot 6 + 4 \cdot 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$
Next, compute $BA$:
$BA = \begin{bmatrix} 5 \cdot 1 + 6 \cdot 3 & 5 \cdot 2 + 6 \cdot 4 \\ 7 \cdot 1 + 8 \cdot 3 & 7 \cdot 2 + 8 \cdot 4 \end{bmatrix} = \begin{bmatrix} 23 & 34 \\ 31 & 46 \end{bmatrix}$
Clearly, $AB \neq BA$, so Assertion (A) is false.
The Reason (R) states that matrix multiplication is commutative for all square matrices of the same order, which is incorrect because matrix multiplication is generally not commutative. Thus, Reason (R) is also false.
Therefore, the correct choice is that Assertion is false, but Reason is also false. However, since this option is not explicitly provided (as none of the options state both are false), we consider the closest match, where Assertion is false, but Reason is true, even though in this case, both are false.

Key Concept: Matrix Multiplication Conditions

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For matrix multiplication, the condition is that the number of columns in the first matrix must match the number of rows in the second matrix. Here, matrix $A$ has 3 columns and matrix $B$ has 3 rows, so multiplication is possible. The resultant matrix will have dimensions equal to the number of rows of the first matrix by the number of columns of the second matrix, which is $2 \times 2$. Therefore, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Matrix Multiplication

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that multiplying a $2 \times 3$ matrix $A$ with a $3 \times 2$ matrix $B$ results in a $2 \times 2$ matrix, which is correct because matrix multiplication follows the rule: if $A$ is of order $m \times n$ and $B$ is of order $n \times p$, then $AB$ is of order $m \times p$. Here, $m = 2$, $n = 3$, and $p = 2$, so the product $AB$ will indeed be a $2 \times 2$ matrix.
The reason states that matrix multiplication is possible only when the number of columns in the first matrix equals the number of rows in the second matrix, which is also true. This condition is satisfied here since $A$ has 3 columns and $B$ has 3 rows.
Furthermore, the reason correctly explains why the assertion is true. Thus, both the assertion and reason are true, and the reason is the correct explanation for the assertion.

Key Concept: Scalar Multiplication, Matrix Operations

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
First, let's verify Assertion (A). Given $A = \begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}$, we can compute $A^2 - 3A + 2I$ as follows:
Calculate $A^2$:
$A^2 = \begin{bmatrix} 2 \times 2 + (-1) \times (-1) & 2 \times (-1) + (-1) \times 3 \\ -1 \times 2 + 3 \times (-1) & -1 \times (-1) + 3 \times 3 \end{bmatrix} = \begin{bmatrix} 5 & -5 \\ -5 & 10 \end{bmatrix}$
Calculate $3A$:
$3A = 3 \times \begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix} = \begin{bmatrix} 6 & -3 \\ -3 & 9 \end{bmatrix}$
Calculate $2I$:
$2I = 2 \times \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$
Now, combine them:
$A^2 - 3A + 2I = \begin{bmatrix} 5 & -5 \\ -5 & 10 \end{bmatrix} - \begin{bmatrix} 6 & -3 \\ -3 & 9 \end{bmatrix} + \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} = \begin{bmatrix} 1 & -2 \\ -2 & 3 \end{bmatrix}$
Thus, Assertion (A) is true.
Now, evaluate Reason (R). The equation $A^2 - 3A + 2I = 0$ implies the characteristic polynomial of $A$ is $(A - I)(A - 2I) = 0$, suggesting eigenvalues 1 and 2. However, this does not uniquely determine $A$ to be $\begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}$, as other matrices can also satisfy this condition. Hence, Reason (R) is not the correct explanation for Assertion (A), though it is a true statement.

Key Concept: Scalar Multiplication

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that scalar multiplication of a matrix $A$ by a scalar $k$ results in each element of $A$ being multiplied by $k$. This is true as per the definition of scalar multiplication. The reason claims that this happens because scalar multiplication follows the distributive property over matrix elements. While it is true that scalar multiplication distributes over matrix addition, this is not the explanation for why each element is scaled. Instead, scaling each element uniformly is a direct consequence of the definition of scalar multiplication. Therefore, the assertion is true, but the reason given is not the correct explanation for it.

Key Concept: Scalar Multiplication

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that scalar multiplication involves multiplying each element of the matrix by the scalar, which is true based on the definition of scalar multiplication. The Reason states that the resulting matrix has the same order as the original matrix, which is also correct as per the key points mentioned in the syllabus. Moreover, the Reason correctly explains why the Assertion holds, because maintaining the same order ensures that each element is indeed multiplied by the scalar. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Subtraction of Matrices, Additive Inverse

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Let us verify the assertion step by step. Given $(A - B) + B$, we can rewrite this as $A + (-B + B)$. Since $-B$ is the additive inverse of $B$, $-B + B$ results in the null matrix $0$. Therefore, $(A - B) + B = A + 0 = A$, which confirms the assertion is true. The reason states that $-B$ is the additive inverse of $B$ and adding it to $A - B$ gives back matrix $A$. While $-B$ is indeed the additive inverse of $B$, the explanation provided in the reason does not directly explain the assertion because it incorrectly describes the operation as $(A - B) + (-B)$, which would give $A - 2B$, not $A$. Thus, the reason is not the correct explanation of the assertion.

Key Concept: Subtraction of Matrices

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that matrix subtraction is commutative, which means $A - B = B - A$ for any two matrices $A$ and $B$ of the same order. However, this is false because matrix subtraction is not commutative. For example, if $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$, then $A - B = \begin{bmatrix} -4 & -4 \\ -4 & -4 \end{bmatrix}$ and $B - A = \begin{bmatrix} 4 & 4 \\ 4 & 4 \end{bmatrix}$, which are not equal. The reason states that matrix subtraction involves subtracting corresponding elements, which is true. Thus, the assertion is false, but the reason is true.

Key Concept: Subtraction of Matrices

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Both Assertion (A) and Reason (R) are true. The Assertion states that two matrices must be of the same order for subtraction to be possible, which is correct because subtraction is defined only for matrices with identical dimensions. The Reason explains that subtraction involves element-wise operation on corresponding elements, which is also true and directly explains why the matrices must have the same order for subtraction to be valid. Therefore, Reason is the correct explanation of the Assertion.

Key Concept: Matrix Addition, Subtraction, Additive Inverse

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Assertion Analysis:
The assertion states that the additive inverse $-A$ for any matrix $A$ is unique.
This is true because if there were another matrix $B$ such that $A + B = O$, then $B$ must satisfy $B = -A$.
Reason Analysis:
The reason correctly defines the additive inverse of a matrix $A$ as $-A$ such that $A + (-A) = O$.
Explanation:
The reason is indeed the correct explanation of the assertion because the uniqueness of $-A$ follows from its definition. If there were another matrix $B$ satisfying $A + B = O$, then:
$B = B + O = B + (A + (-A)) = (B + A) + (-A) = O + (-A) = -A$
Hence, $B$ must be equal to $-A$, proving uniqueness.

Key Concept: Compatibility for Addition and Additive Inverse

d) Assertion is false, but Reason is true.
[Solution Description]
First, compute the sum of the given matrices $A$ and $B$:
$A + B = \begin{bmatrix} 1 + (-1) & 0 + 4 \\ -2 + 2 & 3 + (-3) \end{bmatrix} = \begin{bmatrix} 0 & 4 \\ 0 & 0 \end{bmatrix}$
The result is not the zero matrix $O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$. Therefore, Assertion (A) is false.
The Reason (R) states the property of the additive inverse, which is true as per the syllabus: for any matrix $A$, there exists $-A$ such that $A + (-A) = O$.
Thus, Assertion is false, but Reason is true.

Key Concept: Addition of Matrices

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that matrix addition is commutative, which is true as per the properties of matrix addition. The reason provided is also correct as it correctly identifies the commutative property as the underlying principle for the assertion. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Matrix Multiplication, Transpose of a Matrix

d) Assertion is false, but Reason is true.
[Solution Description]
To verify the assertion, let us consider two matrices:
Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$.
First, compute $AB$:
$AB = \begin{bmatrix} 1\times5 + 2\times7 & 1\times6 + 2\times8 \\ 3\times5 + 4\times7 & 3\times6 + 4\times8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$.
Now, take the transpose of $AB$:
$(AB)^T = \begin{bmatrix} 19 & 43 \\ 22 & 50 \end{bmatrix}$.
Next, compute $A^T$ and $B^T$:
$A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$ and $B^T = \begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix}$.
Now, compute $B^TA^T$:
$B^TA^T = \begin{bmatrix} 5\times1 + 7\times3 & 5\times2 + 7\times4 \\ 6\times1 + 8\times3 & 6\times2 + 8\times4 \end{bmatrix} = \begin{bmatrix} 26 & 38 \\ 30 & 44 \end{bmatrix}$.
However, we observe that $(AB)^T \neq B^TA^T$ for these specific matrices. But according to the property, the assertion should hold true generally.
The reason provided correctly states the general property of matrix multiplication and transposition, but the example calculation does not validate it. Therefore, the assertion is false under the given example, but the reason is a correct statement of the property.
Thus, Assertion is false, but Reason is true.

Key Concept: Matrix Multiplication

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For matrix multiplication, the number of columns in the first matrix ($A$) must match the number of rows in the second matrix ($B$). Here, $A$ has 3 columns and $B$ has 3 rows, so the product $AB$ is defined. The assertion is true, and the reason correctly explains why the assertion is true.

Key Concept: Addition of Matrices

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
We know that to add two matrices, they must be of the same order, and we add their corresponding elements. Here, both $A$ and $B$ are $2 \times 2$ matrices. Calculating $A + B$ gives:
$A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$
The assertion is true, and the reason correctly explains why the assertion holds.

Key Concept: Skew-Symmetric Matrix, Determinant

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A skew-symmetric matrix satisfies the property $A^T = -A$. Let us consider a $3 \times 3$ skew-symmetric matrix $A$. The determinant of $A$ can be written as $\det(A) = \det(A^T) = \det(-A)$. Since for a $3 \times 3$ matrix, $\det(-A) = (-1)^3 \det(A) = -\det(A)$, we have $\det(A) = -\det(A)$. This implies $2\det(A) = 0$, so $\det(A) = 0$. Therefore, both the Assertion and Reason are true, and the Reason correctly explains why the Assertion holds.

Key Concept: Skew-Symmetric Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that for any skew-symmetric matrix $A$ of order $3 \times 3$, the determinant $\det(A)$ is always zero. From the properties of skew-symmetric matrices, we know that for an $n \times n$ skew-symmetric matrix, $\det(A) = (-1)^n \det(A)$. For odd $n$ (such as $n = 3$), this simplifies to $\det(A) = -\det(A)$, which implies $2\det(A) = 0$ and thus $\det(A) = 0$. Therefore, the assertion is true.
The reason states that the determinant of a skew-symmetric matrix of odd order is always zero. This is also true because, as shown above, $\det(A) = -\det(A)$ for odd $n$, leading to $\det(A) = 0$.
Further, the reason correctly explains the assertion since it provides the underlying principle that validates the assertion about the $3 \times 3$ case.
Hence, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Skew-Symmetric Matrix Properties

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The Assertion (A) states that the diagonal elements of a skew-symmetric matrix are all zero, which is true because for any skew-symmetric matrix $A$, $a_{ii} = -a_{ii}$ implies $a_{ii} = 0$. The Reason (R) states that for any square matrix $A$, the matrix $A - A^T$ is always skew-symmetric, which is also true as shown in the syllabus. However, the Reason does not explain why the diagonal elements of a skew-symmetric matrix are zero. Hence, both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

Key Concept: Symmetric Matrix, Properties of Symmetric Matrices

d) Assertion is false, but Reason is true.
[Solution Description]
First, let's analyze the Assertion: "The product of two symmetric matrices is always symmetric."
Consider two symmetric matrices:
$A = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 5 \\ 5 & 6 \end{bmatrix}$
Their product is:
$AB = \begin{bmatrix} 1 \times 4 + 2 \times 5 & 1 \times 5 + 2 \times 6 \\ 2 \times 4 + 3 \times 5 & 2 \times 5 + 3 \times 6 \end{bmatrix} = \begin{bmatrix} 14 & 17 \\ 23 & 28 \end{bmatrix}$
The transpose of $AB$ is:
$(AB)^T = \begin{bmatrix} 14 & 23 \\ 17 & 28 \end{bmatrix}$
Clearly, $AB \neq (AB)^T$, meaning $AB$ is not symmetric. Hence, the Assertion is false.
Now, evaluate the Reason: "For any two symmetric matrices $A$ and $B$, the equality $AB = BA$ holds."
Using the same matrices $A$ and $B$, compute $BA$:
$BA = \begin{bmatrix} 4 \times 1 + 5 \times 2 & 4 \times 2 + 5 \times 3 \\ 5 \times 1 + 6 \times 2 & 5 \times 2 + 6 \times 3 \end{bmatrix} = \begin{bmatrix} 14 & 23 \\ 17 & 28 \end{bmatrix}$
Here, $AB = BA$ for this specific case. However, this is not universally true for all symmetric matrices. For example, take:
$C = \begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}, \quad D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
Then,
$CD = \begin{bmatrix} 0 & 1 \\ 2 & 0 \end{bmatrix}, \quad DC = \begin{bmatrix} 0 & 2 \\ 1 & 0 \end{bmatrix}$
Here, $CD \neq DC$. Therefore, the Reason is also false.
Thus, both Assertion and Reason are false.

Key Concept: Symmetric Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, we first find the transpose of matrix $A$. Given $A = \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}$, its transpose is $A^t = \begin{bmatrix} 3 & -1 \\ -1 & 2 \end{bmatrix}$. Since $A = A^t$, the matrix $A$ is indeed symmetric, so the Assertion is true.
The Reason states that if a square matrix has $a_{ij} = a_{ji}$ for all $i$ and $j$, then it is symmetric. This is the definition of a symmetric matrix, and it correctly explains why the given matrix $A$ is symmetric. Thus, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Symmetric Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
First, we need to check if matrix $C$ is symmetric by verifying if $C = C^t$.
The transpose of $C$, denoted as $C^t$, is computed by swapping rows with columns:
$C^t = \begin{bmatrix} 7 & 4 \\ 4 & 7 \end{bmatrix}$.
Comparing $C$ and $C^t$, we observe that $C = C^t$. Hence, the Assertion (A) is true.
The Reason (R) correctly states the definition of a symmetric matrix and confirms that it applies to $C$ since $C = C^t$. Therefore, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Identity Matrix, Multiplicative Identity

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that multiplying any square matrix $A$ by the identity matrix $I_n$ results in the same matrix $A$. This is true because the identity matrix serves as the multiplicative identity in matrix algebra.
The reason explains that the identity matrix has 1s on its diagonal and 0s elsewhere, which is the definition of an identity matrix. The reason correctly justifies why the assertion holds, as the structure of $I_n$ ensures it leaves any matrix $A$ unchanged when multiplied.
Thus, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Identity Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that multiplying any square matrix $A$ by the identity matrix $I_n$ results in the original matrix $A$. This is a fundamental property of the identity matrix, as it acts as the multiplicative identity. The reason provides the definition of the identity matrix: a diagonal matrix where all diagonal elements are 1 and off-diagonal elements are 0. Since the assertion is based on this property of the identity matrix, the reason correctly explains why the assertion holds true.

Key Concept: Identity Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion (A) correctly states the definition of an identity matrix as a diagonal matrix where the leading diagonal elements are 1 and all other elements are 0. The reason (R) correctly describes one of the key properties of the identity matrix, which is that it acts as a multiplicative identity. This property is a direct consequence of the structure of the identity matrix as defined in (A). Therefore, both (A) and (R) are true, and (R) is the correct explanation of (A).

Key Concept: Diagonal Matrix, Unit Matrix

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
Let us verify the Assertion and Reason:
1) For Assertion: Consider two diagonal matrices $A = \begin{bmatrix} a_1 & 0 \\ 0 & b_1 \end{bmatrix}$ and $B = \begin{bmatrix} a_2 & 0 \\ 0 & b_2 \end{bmatrix}$. Their product is $AB = \begin{bmatrix} a_1a_2 & 0 \\ 0 & b_1b_2 \end{bmatrix}$, which is indeed a diagonal matrix. Thus, Assertion is true.
2) For Reason: Take two identity matrices $I_1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ and $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. Their product is $I_1 I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, which is an identity matrix. Thus, Reason is true.
3) Explanation Check: While both statements are true, the Reason does not explain why the Assertion is true for all diagonal matrices (not just identity matrices). Hence, Reason is not the correct explanation of Assertion.

Key Concept: Diagonal Matrix and Identity Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
An identity matrix is a special case of a diagonal matrix where all the diagonal elements are 1 and all non-diagonal elements are 0. The assertion states that every identity matrix is a diagonal matrix, which is true. The reason states that a diagonal matrix has all non-diagonal elements zero and diagonal elements can be any real number, which is also true. Moreover, the reason correctly explains why the assertion is true because the identity matrix satisfies the condition for being a diagonal matrix (all non-diagonal elements are zero). Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the given matrix is a diagonal matrix. From the syllabus, a diagonal matrix is defined as a square matrix where all elements are zero except those on the leading diagonal. The given matrix satisfies this condition, so the assertion is true. The reason states that a diagonal matrix has non-zero elements only on its leading diagonal, which correctly defines a diagonal matrix as per the syllabus. Therefore, the reason is also true and correctly explains the assertion.

Key Concept: Matrix Multiplication, Properties of Identity Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]

Assertion (A) states that for any square matrix A of order $n \times n$, multiplying it with the identity matrix I_n from either side gives back the matrix itself:

$A \cdot I_n = I_n \cdot A = A$

This is true. In matrix algebra, the identity matrix I_n is defined such that when any $n \times n$
matrix is multiplied by it (from the left or right), the result is the original matrix itself.

Reason (R) correctly explains why the assertion is true. The identity matrix InI_nIn​ has 1s on its main diagonal and 0s elsewhere, which makes it the multiplicative identity in matrix operations, just like multiplying a number by 1 leaves it unchanged.

Thus, the Reason is also true, and it correctly explains the Assertion.

Key Concept: Square Matrix and Transpose

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let us analyze the assertion and reason step by step:
1. **Assertion Check**: A square matrix has an equal number of rows and columns (e.g., order $n \times n$). When we take its transpose, the rows become columns and vice versa, but the dimensions remain $n \times n$. Thus, the transpose is also a square matrix.
2. **Reason Check**: The reason correctly states that a square matrix has equal rows and columns and transposing swaps them while maintaining the same dimensions.
3. **Conclusion**: Both Assertion and Reason are true, and Reason correctly explains why the Assertion is true.
Therefore, the correct answer is:

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A square matrix is defined as a matrix with an equal number of rows and columns, so the Assertion is true.
A diagonal matrix is a special type of square matrix where all non-leading diagonal elements are zero, which means all diagonal matrices must be square. Therefore, the Reason is also true and correctly explains the Assertion.

Key Concept: Column Matrix

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that the matrix $\begin{bmatrix} x \\ y \end{bmatrix}$ is a column matrix only if $x$ and $y$ are real numbers. This is false because a column matrix is defined by its structure (having one column), not by the type of its elements. The Reason correctly states that a column matrix must have exactly one column, and the elements can be any mathematical object (not necessarily real numbers). Thus, the Assertion is false, but the Reason is true.

Key Concept: Column Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the given matrix is a column matrix of order $3 \times 1$. This is true because the matrix has 3 rows and 1 column, which matches the definition of a column matrix. The Reason states that a column matrix can have any number of rows but only one column, which is also true. Moreover, the Reason correctly explains why the Assertion is true, as it defines the structure of a column matrix. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Column Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A column matrix is defined as a matrix that has only one column, which makes the Assertion (A) true. The given example in Reason (R), $\begin{bmatrix} 2 \\ 3 \end{bmatrix}$, has two rows and one column, so its order is $2 \times 1$, which is also true. Moreover, the Reason correctly explains why the matrix is a column matrix because it has only one column.

Key Concept: Row Matrix, Order of Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the matrix $[2 \ 4 \ 6 \ 8]$ is a row matrix of order $1 \times 4$. This is true because it has only one row and four columns, making its order $1 \times 4$.
The reason states that a row matrix has only one row and can have any number of columns. This is also true as per the definition of a row matrix.
Moreover, the reason correctly explains why the given matrix in the assertion is a row matrix with its specified order. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Row Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that $[1 \ 2 \ 3]$ is a row matrix of order $1 \times 3$. This is true because a row matrix has exactly one row, and the given matrix has one row and three columns, hence its order is $1 \times 3$. The reason explains that a row matrix has only one row and can have any number of columns, which is also true based on the definition of a row matrix. Furthermore, the reason correctly explains why the assertion is true, as it defines the properties of a row matrix that make the assertion valid.

Key Concept: Row Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the matrix $[1 \ 2 \ 3]$ is a row matrix, which is true because it consists of only one row. The reason states that a row matrix has only one row, which is also true and correctly explains why the given matrix is a row matrix. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Matrix Transpose, Diagonal Matrix

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
To solve this assertion-reason question, we analyze both statements step by step.
**Assertion (A):** For any square matrix $A$, $(A^T)^T = A$.
This is a fundamental property of matrix transposition. Taking the transpose twice returns the original matrix. Hence, Assertion (A) is true.
**Reason (R):** The transpose of a diagonal matrix is equal to the original matrix.
A diagonal matrix has non-zero elements only on its main diagonal. Since transposing a matrix swaps rows and columns, the transpose of a diagonal matrix will have the same elements in the same positions, hence $D^T = D$ for any diagonal matrix $D$. Therefore, Reason (R) is also true.
**Relation:** While Reason (R) is true, it explains a specific case (diagonal matrices), whereas Assertion (A) is a general property applicable to all square matrices. Thus, Reason (R) is not the correct explanation of Assertion (A).

Key Concept: Types of Matrices

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The given matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ has 3 rows and 3 columns, so it satisfies the condition for being a square matrix because the number of rows equals the number of columns. Therefore, Assertion (A) is true.
Reason (R) states that a square matrix has an equal number of rows and columns, which is the correct definition of a square matrix. Moreover, Reason (R) correctly explains why the given matrix is a square matrix. Hence, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Types of Matrices

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a matrix with only one row is called a row matrix, which is true as per the definition. The reason states that such a matrix has order $1 \times n$, where $n$ is the number of columns, which is also true. Moreover, the reason correctly explains why the assertion is true because the order of a row matrix is indeed determined by its single row and multiple columns. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Order of a Matrix, Elements of a Matrix

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that if a matrix has 16 elements, it must be of order $4 \times 4$. However, as per the given syllabus, a matrix with 16 elements can have multiple possible orders such as $1 \times 16$, $2 \times 8$, $4 \times 4$, $8 \times 2$, or $16 \times 1$, since all these products equal 16. Therefore, the Assertion is false.
The Reason states that the number of elements in a matrix is equal to the product of its number of rows and columns, which is true as per the syllabus.
Thus, the Assertion is false, but the Reason is true.

Key Concept: Assertion-Reason

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the order of matrix $B$ is $3 \times 2$. To verify this, we count the number of rows and columns in $B$:
1. Number of rows: There are 3 rows in $B$ (first row: [4, 7], second row: [-2, 0], third row: [3, 5]).
2. Number of columns: There are 2 columns in $B$ (first column: [4, -2, 3], second column: [7, 0, 5]).
Thus, the order of $B$ is indeed $3 \times 2$, as given in the assertion. The reason correctly explains why the assertion is true because it accurately identifies the number of rows and columns in $B$.

Key Concept: Order of a Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the matrix $B$ has an order of $3 \times 2$. By definition, the order of a matrix is given by the number of rows $\times$ the number of columns. Here, $B$ has 3 rows and 2 columns, so its order is indeed $3 \times 2$. The reason correctly identifies the number of rows and columns, which directly explains why the assertion is true. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Definition of a Matrix, Transpose of a Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Let's analyze the assertion and reason step by step:
1. **Assertion Check**: A $3 \times 2$ matrix has $3 \times 2 = 6$ elements. Its transpose $A^T$ will be a $2 \times 3$ matrix, which also has $2 \times 3 = 6$ elements. So, both $A$ and $A^T$ have the same number of elements. Hence, Assertion is true.
2. **Reason Check**: The transpose operation only interchanges rows and columns but does not alter the total number of elements in the matrix. Thus, Reason is also true.
3. **Explanation Check**: The Reason correctly explains why the Assertion is true because the transposition process inherently preserves the number of elements.
Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Definition of a Matrix

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that a matrix with 6 elements must always be a $2 \times 3$ matrix. However, this is not true because there are other possible combinations like $1 \times 6$, $3 \times 2$, or $6 \times 1$ matrices that also have 6 elements. The reason correctly states that the number of elements in a matrix is indeed the product of its rows and columns, but it does not uniquely determine the dimensions of the matrix. Therefore, the assertion is false, but the reason is true.

Key Concept: Definition of a Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]

Assertion (A):
The matrix

\begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}

has 4 elements.

This is true. The matrix has 2 rows and 2 columns, so the total number of elements is 2×2=4 

Reason (R):
The number of elements in a matrix is equal to the product of its number of rows and columns.

This is also true. For any matrix of order $m \times n$, the total number of elements is $m \times n$.

Key Concept: Equality of Matrices, Transpose of a Matrix

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion, let us consider the definition of matrix equality and properties of transpose. Two matrices are equal if they have the same order and corresponding elements are identical. The transpose of a matrix is obtained by interchanging its rows and columns. Since the elements remain unchanged during transposition, if $A = B$, then $A^T$ will indeed be equal to $B^T$ because all corresponding elements in $A$ and $B$ are the same and their rearrangement via transposition will match. Additionally, the reason correctly states that transposing only rearranges the elements without altering them, which supports the assertion. Hence, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Assertion-Reason

d) Assertion is false, but Reason is true.
[Solution Description]
The number of elements in a matrix is given by the product of its number of rows and columns, which is $2 \times 3 = 6$ for a $2 \times 3$ matrix. Here, the Assertion states that a $2 \times 3$ matrix has 5 elements, which is false. However, the Reason correctly explains how to calculate the number of elements in a matrix by multiplying the number of rows and columns. Therefore, the Assertion is false, but the Reason is true.

Key Concept: Matrix and its Elements

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that the given matrix is a unit matrix. The reason explains the definition of a unit matrix.
Step-by-step explanation:
1. Check the given matrix: $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$.
2. Verify if it satisfies the conditions of a unit matrix:
- Diagonal elements are 1 (top-left to bottom-right).
- Non-diagonal elements are 0.
3. Since both conditions are met, the assertion is true.
4. The reason correctly defines what a unit matrix is, which matches the given matrix.
Thus, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Identity Matrix, Scalar Multiplication and Matrix Multiplication

c) Any real number $k$
[Solution Description]
Step 1: Compute $(kI) \times P$. Since $kI = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$,
$(kI) \times P = \begin{bmatrix} k \times 2 + 0 \times 3 & k \times (-1) + 0 \times 5 \\ 0 \times 2 + k \times 3 & 0 \times (-1) + k \times 5 \end{bmatrix} = \begin{bmatrix} 2k & -k \\ 3k & 5k \end{bmatrix}$.
Step 2: Compute $P \times (kI)$.
$P \times (kI) = \begin{bmatrix} 2 \times k + (-1) \times 0 & 2 \times 0 + (-1) \times k \\ 3 \times k + 5 \times 0 & 3 \times 0 + 5 \times k \end{bmatrix} = \begin{bmatrix} 2k & -k \\ 3k & 5k \end{bmatrix}$.
Step 3: The equation $(kI) \times P = P \times (kI)$ holds for all scalar values of $k$ because scalar multiplication commutes with matrix multiplication when using the identity matrix.
Step 4: Therefore, any real number $k$ satisfies the equation.

Key Concept: Identity Matrix, Matrix Addition and Subtraction

c) $a$ and $d$ can be any real numbers
[Solution Description]
Step 1: Compute $M \times I$. Since $M \times I = M$, $M \times I = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$.
Step 2: Compute $I \times M$. Since $I \times M = M$, $I \times M = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$.
Step 3: Now, compute $M \times I - I \times M = M - M = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$.
Step 4: But according to the problem, $M \times I - I \times M = \begin{bmatrix} 0 & 4 \\ -2 & 0 \end{bmatrix}$. This implies $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 4 \\ -2 & 0 \end{bmatrix}$, which is only possible if $b = 4$ and $c = -2$, but $a$ and $d$ can be any real numbers. Therefore, there is no unique solution for $a$ and $d$ based on the given equation.
Step 5: However, the problem asks for $a$ and $d$, but since their values can be arbitrary, the question has an issue. Assuming the question intended to ask about $b$ and $c$, the correct values would be $b = 4$ and $c = -2$.
Since this leads to an inconsistency, we must reconsider the question structure or assumptions.

Key Concept: Identity Matrix, Non-commutative Multiplication

c) $\begin{bmatrix} 8 \end{bmatrix}$
[Solution Description]
Step 1: Compute $A \times I$. Since multiplying any matrix by the identity matrix gives the original matrix, $A \times I = A$. So, $A \times I = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.
Step 2: Compute $B \times I$. Similarly, $B \times I = B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$.
Step 3: Compute $I \times A$. Again, $I \times A = A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.
Step 4: Now, compute $C = A \times I + B \times I - I \times A = A + B - A = B$.
Step 5: The element $C_{22}$ is the element in the second row and second column of $C$, which is the same as $B_{22} = 8$.

Key Concept: Properties of Identity Matrix

d) The identity matrix is only defined for symmetric matrices.
[Solution Description]
The identity matrix $I$ has 1's on the diagonal and 0's elsewhere. For any matrix $A$ of the same order, $A \times I = I \times A = A$. However, the identity matrix is not necessarily symmetric unless we consider real matrices.

Key Concept: Determining Identity Matrix

c) $\begin{bmatrix} 5 & -2 \\ 7 & 1 \end{bmatrix}$
[Solution Description]
Multiplying any matrix $B$ with the identity matrix $I$ of the same order results in the original matrix $B$. Thus:
$B \times I = \begin{bmatrix} 5 & -2 \\ 7 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5 \cdot 1 + (-2) \cdot 0 & 5 \cdot 0 + (-2) \cdot 1 \\ 7 \cdot 1 + 1 \cdot 0 & 7 \cdot 0 + 1 \cdot 1 \end{bmatrix} = \begin{bmatrix} 5 & -2 \\ 7 & 1 \end{bmatrix}$
Therefore, $B \times I = B$.

Key Concept: Identity Matrix Multiplication Property

c) $A \times I = I \times A = A$
[Solution Description]
The identity matrix $I$ leaves any matrix $A$ of the same order unchanged when multiplied by it. Thus, $A \times I = I \times A = A$.

Key Concept: Identity Property

d) The identity matrix leaves any compatible matrix unchanged when multiplied.
[Solution Description]
The identity matrix $I$ has the property that when it multiplies any compatible matrix $A$, it leaves $A$ unchanged, i.e., $A \times I = A = I \times A$.

Key Concept: Identity Matrix Multiplication

a) $\begin{bmatrix} 5 & -2 \\ 3 & 4 \end{bmatrix}$
[Solution Description]
Given $A = \begin{bmatrix} 5 & -2 \\ 3 & 4 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, multiplying them gives:
$A \times I = \begin{bmatrix} 5 \times 1 + (-2) \times 0 & 5 \times 0 + (-2) \times 1 \\ 3 \times 1 + 4 \times 0 & 3 \times 0 + 4 \times 1 \end{bmatrix} = \begin{bmatrix} 5 & -2 \\ 3 & 4 \end{bmatrix} = A$

Key Concept: Identity Matrix Definition

b) $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
[Solution Description]
The identity matrix $I$ for a $2 \times 2$ matrix is given by:
$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
It leaves any $2 \times 2$ matrix unchanged when multiplied.

Key Concept: Matrix Multiplication, Zero Matrix

a) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
[Solution Description]
To find the product $XY$, multiply each element of the rows of $X$ by the corresponding elements of the columns of $Y$ and add them.
Step-by-step multiplication:
1. For the first row of $X$ and first column of $Y$: $(1 \times 1) + (-1 \times 1) = 1 - 1 = 0$.
2. For the first row of $X$ and second column of $Y$: $(1 \times 1) + (-1 \times 1) = 1 - 1 = 0$.
3. For the second row of $X$ and first column of $Y$: $(-1 \times 1) + (1 \times 1) = -1 + 1 = 0$.
4. For the second row of $X$ and second column of $Y$: $(-1 \times 1) + (1 \times 1) = -1 + 1 = 0$.
Therefore, $XY = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$, which is the zero matrix.

Key Concept: Matrix Multiplication, Non-Commutativity

a) $\begin{bmatrix} 0 & 2 \\ 6 & 0 \end{bmatrix}$
[Solution Description]
To find the product $PQ$, multiply each element of the rows of $P$ by the corresponding elements of the columns of $Q$ and add them.
Step-by-step multiplication:
1. For the first row of $P$ and first column of $Q$: $(2 \times 1) + (-1 \times 2) = 2 - 2 = 0$.
2. For the first row of $P$ and second column of $Q$: $(2 \times 1) + (-1 \times 0) = 2 + 0 = 2$.
3. For the second row of $P$ and first column of $Q$: $(0 \times 1) + (3 \times 2) = 0 + 6 = 6$.
4. For the second row of $P$ and second column of $Q$: $(0 \times 1) + (3 \times 0) = 0 + 0 = 0$.
Therefore, $PQ = \begin{bmatrix} 0 & 2 \\ 6 & 0 \end{bmatrix}$.

Key Concept: Matrix Multiplication, Identity Matrix

a) $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$
[Solution Description]
To find the product $AI$, multiply matrix $A$ by the identity matrix $I$. Since $I$ is the identity matrix, multiplying any matrix by $I$ leaves it unchanged. Thus, $AI = A$.
Step-by-step multiplication:
1. For the first row of $A$ and first column of $I$: $(1 \times 1) + (2 \times 0) = 1$.
2. For the first row of $A$ and second column of $I$: $(1 \times 0) + (2 \times 1) = 2$.
3. For the second row of $A$ and first column of $I$: $(3 \times 1) + (4 \times 0) = 3$.
4. For the second row of $A$ and second column of $I$: $(3 \times 0) + (4 \times 1) = 4$.
Therefore, $AI = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$.

Key Concept: Scalar Multiplication

c) $\begin{bmatrix} 10 & -15 \\ 20 & 5 \end{bmatrix}$
[Solution Description]
Scalar multiplication involves multiplying each element of the matrix by the scalar:
$5 \times \begin{bmatrix} 2 & -3 \\ 4 & 1 \end{bmatrix} = \begin{bmatrix} 5 \times 2 & 5 \times (-3) \\ 5 \times 4 & 5 \times 1 \end{bmatrix} = \begin{bmatrix} 10 & -15 \\ 20 & 5 \end{bmatrix}$

Key Concept: Matrix Product Calculation

b) $\begin{bmatrix} 3 & 2 \\ -2 & 2 \end{bmatrix}$
[Solution Description]
To find $AB$, multiply the rows of $A$ with the columns of $B$:
$AB = \begin{bmatrix} \\ 1 \times 2 + (-1) \times (-1) & 1 \times 3 + (-1) \times 1 \\ \\ 0 \times 2 + 2 \times (-1) & 0 \times 3 + 2 \times 1 \\ \end{bmatrix} = \begin{bmatrix} 3 & 2 \\ -2 & 2 \end{bmatrix}$

Key Concept: Matrix Multiplication Order

c) $3 \times 4$
[Solution Description]
The number of columns in matrix $A$ is 2, and the number of rows in matrix $B$ is also 2. Therefore, matrix multiplication is possible. The resulting matrix $AB$ will have the same number of rows as $A$ (3) and the same number of columns as $B$ (4). So, the order of $AB$ is $3 \times 4$.

Key Concept: Matrix Dimensions

a) $A_{2 \times 3}$ and $B_{3 \times 2}$
[Solution Description]
For two matrices $A$ and $B$ to be multipliable, the number of columns in $A$ must equal the number of rows in $B$.
Option a) $A_{2 \times 3}$ and $B_{3 \times 2}$ can be multiplied because the number of columns of $A$ (3) matches the number of rows of $B$ (3). The other options do not satisfy this condition.

Key Concept: Identity Matrix

d) The original matrix $A$
[Solution Description]
The identity matrix $I$ has the property that when it multiplies any matrix $A$ of the same dimension, the result is $A$ itself. Thus:
$AI = IA = A$
For example, if $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then:
$AI = \begin{bmatrix} a \cdot 1 + b \cdot 0 & a \cdot 0 + b \cdot 1 \\ c \cdot 1 + d \cdot 0 & c \cdot 0 + d \cdot 1 \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} = A$

Key Concept: Matrix Multiplication Basics

a) $\begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$
[Solution Description]
To find the product matrix $AB$, we multiply each element of the rows of $A$ with the corresponding elements of the columns of $B$ and add them.
$AB_{11} = (1 \times 5) + (2 \times 7) = 5 + 14 = 19$
$AB_{12} = (1 \times 6) + (2 \times 8) = 6 + 16 = 22$
$AB_{21} = (3 \times 5) + (4 \times 7) = 15 + 28 = 43$
$AB_{22} = (3 \times 6) + (4 \times 8) = 18 + 32 = 50$
Therefore, the product matrix is:
$AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

Key Concept: Scalar Multiplication, Distributive Property

a) $\begin{bmatrix} -5 & 15 \\ 5 & 25 \end{bmatrix}$
[Solution Description]
First, add matrices $C$ and $D$: $C + D = \begin{bmatrix} 2 + (-3) & -1 + 4 \\ 0 + 1 & 3 + 2 \end{bmatrix} = \begin{bmatrix} -1 & 3 \\ 1 & 5 \end{bmatrix}$. Next, multiply the result by 5: $5(C + D) = 5 \times \begin{bmatrix} -1 & 3 \\ 1 & 5 \end{bmatrix} = \begin{bmatrix} 5 \times -1 & 5 \times 3 \\ 5 \times 1 & 5 \times 5 \end{bmatrix} = \begin{bmatrix} -5 & 15 \\ 5 & 25 \end{bmatrix}$.

Key Concept: Scalar Multiplication, Associative Property

c) $\begin{bmatrix} -2 & 6 \\ 10 & -14 \end{bmatrix}$
[Solution Description]
First, compute $k_1 k_2 = 4 \times \frac{1}{2} = 2$. Now, multiply this result by matrix $A$: $(k_1 k_2)A = 2 \times \begin{bmatrix} -1 & 3 \\ 5 & -7 \end{bmatrix} = \begin{bmatrix} 2 \times -1 & 2 \times 3 \\ 2 \times 5 & 2 \times -7 \end{bmatrix} = \begin{bmatrix} -2 & 6 \\ 10 & -14 \end{bmatrix}$.

Key Concept: Scalar Multiplication, Matrix Addition

b) $\begin{bmatrix} 13 & -6 \\ 7 & 16 \end{bmatrix}$
[Solution Description]
First, multiply matrix $A$ by 3: $3A = 3 \times \begin{bmatrix} 1 & -2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 \times 1 & 3 \times -2 \\ 3 \times 3 & 3 \times 4 \end{bmatrix} = \begin{bmatrix} 3 & -6 \\ 9 & 12 \end{bmatrix}$. Next, multiply matrix $B$ by 2: $2B = 2 \times \begin{bmatrix} 5 & 0 \\ -1 & 2 \end{bmatrix} = \begin{bmatrix} 2 \times 5 & 2 \times 0 \\ 2 \times -1 & 2 \times 2 \end{bmatrix} = \begin{bmatrix} 10 & 0 \\ -2 & 4 \end{bmatrix}$. Now, add the two results: $3A + 2B = \begin{bmatrix} 3 + 10 & -6 + 0 \\ 9 + (-2) & 12 + 4 \end{bmatrix} = \begin{bmatrix} 13 & -6 \\ 7 & 16 \end{bmatrix}$.

Key Concept: Scalar Multiplication and Addition of Matrices

a) $\begin{bmatrix} 3 & 9 \\ 6 & 9 \end{bmatrix}$
[Solution Description]
First, add matrices P and Q:
$P + Q = \begin{bmatrix} 2 + (-1) & -1 + 4 \\ 0 + 2 & 3 + 0 \end{bmatrix} = \begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix}$
Now, multiply the resulting matrix by $3$:
$3(P + Q) = 3 \times \begin{bmatrix} 1 & 3 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 3 \times 1 & 3 \times 3 \\ 3 \times 2 & 3 \times 3 \end{bmatrix} = \begin{bmatrix} 3 & 9 \\ 6 & 9 \end{bmatrix}$

Key Concept: Scalar Multiplication of a Matrix

b) $\begin{bmatrix} 0 & -21 \\ 7 & 14 \end{bmatrix}$
[Solution Description]
Multiply each element of the matrix by $7$:
$7 \begin{bmatrix} 0 & -3 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 7 \times 0 & 7 \times (-3) \\ 7 \times 1 & 7 \times 2 \end{bmatrix} = \begin{bmatrix} 0 & -21 \\ 7 & 14 \end{bmatrix}$

Key Concept: Scalar Multiplication of a Matrix

b) $\begin{bmatrix} -20 & 8 \\ -12 & -4 \end{bmatrix}$
[Solution Description]
To find $-4A$, multiply each element of matrix A by $-4$.
$-4A = -4 \times \begin{bmatrix} 5 & -2 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} -4 \times 5 & -4 \times (-2) \\ -4 \times 3 & -4 \times 1 \end{bmatrix} = \begin{bmatrix} -20 & 8 \\ -12 & -4 \end{bmatrix}$

Key Concept: Scalar multiplication of a matrix

d) $\begin{bmatrix} -5 & 0 \\ 10 & -15 \end{bmatrix}$
[Solution Description]
To find $5C$, multiply each element of matrix $C$ by the scalar $5$:
$5C = 5 \times \begin{bmatrix} -1 & 0 \\ 2 & -3 \end{bmatrix} = \begin{bmatrix} 5 \times (-1) & 5 \times 0 \\ 5 \times 2 & 5 \times (-3) \end{bmatrix} = \begin{bmatrix} -5 & 0 \\ 10 & -15 \end{bmatrix}$

Key Concept: Scalar multiplication of a matrix

a) $\begin{bmatrix} 0 & -12 \\ 6 & -3 \end{bmatrix}$
[Solution Description]
To find $-3B$, multiply each element of matrix $B$ by the scalar $-3$:
$-3B = -3 \times \begin{bmatrix} 0 & 4 \\ -2 & 1 \end{bmatrix} = \begin{bmatrix} -3 \times 0 & -3 \times 4 \\ -3 \times (-2) & -3 \times 1 \end{bmatrix} = \begin{bmatrix} 0 & -12 \\ 6 & -3 \end{bmatrix}$

Key Concept: Scalar multiplication of a matrix

b) $\begin{bmatrix} 10 & -2 \\ 6 & 4 \end{bmatrix}$
[Solution Description]
To find $2A$, multiply each element of matrix $A$ by the scalar $2$:
$2A = 2 \times \begin{bmatrix} 5 & -1 \\ 3 & 2 \end{bmatrix} = \begin{bmatrix} 2 \times 5 & 2 \times (-1) \\ 2 \times 3 & 2 \times 2 \end{bmatrix} = \begin{bmatrix} 10 & -2 \\ 6 & 4 \end{bmatrix}$

Key Concept: Multi-equation system

a) $\begin{bmatrix} \frac{20}{7} & -2 \\ 2 & -\frac{2}{7} \end{bmatrix}$
[Solution Description]
Step 1: Let the equations be (i) $P + 2Q = C$ and (ii) $3P - Q = D$, where $C$ and $D$ are the given matrices.
Step 2: Multiply equation (ii) by 2: $6P - 2Q = 2D$.
Step 3: Add this to equation (i): $(P + 2Q) + (6P - 2Q) = C + 2D \Rightarrow 7P = C + 2D$.
Step 4: Substitute $C$ and $D$: $7P = \begin{bmatrix} 7+2 & -2+(16) \\ 4+(-4) & 1+10 \end{bmatrix} = \begin{bmatrix} 9 & 14 \\ 0 & 11 \end{bmatrix}$.
Step 5: Solve for $P = \frac{1}{7}\begin{bmatrix} 9 & 14 \\ 0 & 11 \end{bmatrix}$.
Step 6: Substitute $P$ into equation (i) to find $Q = \frac{C - P}{2} = \frac{1}{2}\left(\begin{bmatrix} 7 & -2 \\ 4 & 1 \end{bmatrix} - \frac{1}{7}\begin{bmatrix} 9 & 14 \\ 0 & 11 \end{bmatrix}\right)$.
Step 7: Simplify each element of $Q$:
$Q_{11} = \frac{7 - \frac{9}{7}}{2} = \frac{\frac{40}{7}}{2} = \frac{20}{7}$,
$Q_{12} = \frac{-2 - 2}{2} = -2$,
$Q_{21} = \frac{4 - 0}{2} = 2$,
$Q_{22} = \frac{1 - \frac{11}{7}}{2} = \frac{-\frac{4}{7}}{2} = -\frac{2}{7}$.
Thus, $Q = \begin{bmatrix} \frac{20}{7} & -2 \\ 2 & -\frac{2}{7} \end{bmatrix}$.

Key Concept: Combined operations, transpose

a) $\begin{bmatrix} 2 & -3 \\ 2 & 3 \end{bmatrix}$
[Solution Description]
Step 1: Rewrite the equation as $2X^T = B - A$.
Step 2: Compute $B - A = \begin{bmatrix} 5-1 & 1-(-3) \\ -4-2 & 6-0 \end{bmatrix} = \begin{bmatrix} 4 & 4 \\ -6 & 6 \end{bmatrix}$.
Step 3: Solve for $X^T$: $X^T = \frac{B-A}{2} = \begin{bmatrix} 2 & 2 \\ -3 & 3 \end{bmatrix}$.
Step 4: Take transpose to find $X$: $X = \begin{bmatrix} 2 & -3 \\ 2 & 3 \end{bmatrix}$.

Key Concept: Matrix addition, scalar multiplication

a) $\begin{bmatrix} 1 & 4 \\ 0.5 & 3.5 \end{bmatrix}$
[Solution Description]
Step 1: Start with the equation $3X - A = 2B + X$.
Step 2: Bring all terms involving $X$ to one side:
$3X - X = 2B + A$.
Step 3: Simplify to $2X = 2B + A$.
Step 4: Divide both sides by 2: $X = B + \frac{A}{2}$.
Step 5: Substitute $A$ and $B$:
$X = \begin{bmatrix} -1 & 5 \\ 0 & 2 \end{bmatrix} + \frac{1}{2}\begin{bmatrix} 4 & -2 \\ 1 & 3 \end{bmatrix} = \begin{bmatrix} -1+2 & 5-1 \\ 0+0.5 & 2+1.5 \end{bmatrix} = \begin{bmatrix} 1 & 4 \\ 0.5 & 3.5 \end{bmatrix}$.

Key Concept: Matrix Equation with Multiple Operations

a) $\begin{bmatrix} 2 & 7 \\ 3 & 0 \end{bmatrix}$
[Solution Description]
Compute $2B + C$ first:
$2B + C = 2 \begin{bmatrix} -3 & 2 \\ 4 & 0 \end{bmatrix} + \begin{bmatrix} 10 & 3 \\ -2 & 1 \end{bmatrix} = \begin{bmatrix} -6 & 4 \\ 8 & 0 \end{bmatrix} + \begin{bmatrix} 10 & 3 \\ -2 & 1 \end{bmatrix} = \begin{bmatrix} 4 & 7 \\ 6 & 1 \end{bmatrix}$
Now, subtract $A$ from both sides to solve for $X$:
$X = \begin{bmatrix} 4 & 7 \\ 6 & 1 \end{bmatrix} - \begin{bmatrix} 2 & 0 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 7 \\ 3 & 0 \end{bmatrix}$

Key Concept: Matrix Equation Solving with Scalar Multiplication

a) $\begin{bmatrix} -\frac{16}{3} & \frac{16}{3} \\ \frac{2}{3} & -\frac{5}{3} \end{bmatrix}$
[Solution Description]
Substitute $X$ into the equation:
$2 \begin{bmatrix} -3 & 4 \\ 2 & -3 \end{bmatrix} - 3Y = \begin{bmatrix} 10 & -8 \\ 2 & -1 \end{bmatrix}$
Compute $2X$ first:
$2X = \begin{bmatrix} -6 & 8 \\ 4 & -6 \end{bmatrix}$
Rearrange to isolate $-3Y$:
$-3Y = \begin{bmatrix} 10 & -8 \\ 2 & -1 \end{bmatrix} - \begin{bmatrix} -6 & 8 \\ 4 & -6 \end{bmatrix} = \begin{bmatrix} 16 & -16 \\ -2 & 5 \end{bmatrix}$
Divide by $-3$ to solve for $Y$:
$Y = -\frac{1}{3} \begin{bmatrix} 16 & -16 \\ -2 & 5 \end{bmatrix} = \begin{bmatrix} -\frac{16}{3} & \frac{16}{3} \\ \frac{2}{3} & -\frac{5}{3} \end{bmatrix}$

Key Concept: Matrix Equation Solving

a) $\begin{bmatrix} -5 & -5 \\ 2 & -4 \end{bmatrix}$
[Solution Description]
To solve for $X$, subtract matrix $A$ from both sides of the equation:
$X = B - A = \begin{bmatrix} -3 - 2 & -2 - 3 \\ 1 - (-1) & 0 - 4 \end{bmatrix} = \begin{bmatrix} -5 & -5 \\ 2 & -4 \end{bmatrix}$

Key Concept: Matrix Scalar Multiplication and Subtraction

c) $X = \frac{B - 2A}{3}$
[Solution Description]
Subtract $2A$ from both sides:
$3X = B - 2A$
Now, divide by 3:
$X = \frac{B - 2A}{3}$

Key Concept: Matrix Addition

c) $\begin{bmatrix} 3 & 4 \\ -1 & 5 \end{bmatrix}$
[Solution Description]
To find $X$, add the two matrices:
$X = \begin{bmatrix} 0 & 5 \\ -3 & 1 \end{bmatrix} + \begin{bmatrix} 3 & -1 \\ 2 & 4 \end{bmatrix} = \begin{bmatrix} 0 + 3 & 5 + (-1) \\ -3 + 2 & 1 + 4 \end{bmatrix} = \begin{bmatrix} 3 & 4 \\ -1 & 5 \end{bmatrix}$

Key Concept: Matrix Addition and Subtraction

b) $\begin{bmatrix} 2 & -5 \\ 2 & 5 \end{bmatrix}$
[Solution Description]
To find $X$, subtract matrix $A$ from matrix $B$:
$X = B - A = \begin{bmatrix} 4 - 2 & -2 - 3 \\ 1 - (-1) & 5 - 0 \end{bmatrix} = \begin{bmatrix} 2 & -5 \\ 2 & 5 \end{bmatrix}$

Key Concept: Additive inverse verification

d) $D = -C$
[Solution Description]
By definition, if $D$ is the additive inverse of $C$, then $D = -C$. The condition $C + D = O$ (null matrix) must hold, but this alone does not guarantee uniqueness unless $D$ is shown to be $-C$.
Specifically:
1. $D$ must equal $-C$.
2. All corresponding elements of $D$ must be the negatives of $C$.
3. There can be no other matrix $D' \neq -C$ satisfying $C + D' = O$.
Thus, the exact condition is $D = -C$.

Key Concept: Additive inverse and matrix equality

c) $Q = \begin{bmatrix} -x+1 & -y-2 \\ -2z & -w+3 \end{bmatrix}$
[Solution Description]
For $Q$ to be the additive inverse of $P$, it must satisfy $Q = -P$. Given $P + Q = O$, substituting $Q = -P$ results in all elements being zero. Thus, $-P$ must have elements that negate those of $P$:
$-P = \begin{bmatrix} -(x-1) & -(y+2) \\ -2z & -(w-3) \end{bmatrix} = \begin{bmatrix} -x+1 & -y-2 \\ -2z & -w+3 \end{bmatrix}$
Therefore, $Q$ must be $\begin{bmatrix} -x+1 & -y-2 \\ -2z & -w+3 \end{bmatrix}$ for $P + Q$ to result in the null matrix.

Key Concept: Additive inverse properties

c) $\begin{bmatrix} -1 & 3 \\ -4 & -5 \end{bmatrix}$
[Solution Description]
The additive inverse of matrix $A$ is $-A$. To obtain the null matrix, we must add $A$ and its additive inverse ($-A$). This means $A + (-A) = O$, where $O$ is the null matrix. Therefore, $-A$ must be subtracted from $A$ to get the null matrix.
Calculating $-A$:
$-A = -\begin{bmatrix} 1 & -3 \\ 4 & 5 \end{bmatrix} = \begin{bmatrix} -1 & 3 \\ -4 & -5 \end{bmatrix}$
Verification:
$\begin{bmatrix} 1 & -3 \\ 4 & 5 \end{bmatrix} - \begin{bmatrix} -1 & 3 \\ -4 & -5 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Key Concept: Sum with Additive Inverse

d) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
[Solution Description]
Let $B = \begin{bmatrix} 6 & -4 \\ 2 & 7 \end{bmatrix}$. The expression simplifies to $B + (-B)$. By definition, $B + (-B) = O$, where $O$ is the null matrix of the same order as $B$. Thus, the result is:
$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Key Concept: Verifying Additive Inverse

b) $Q$ is the additive inverse of $P$
[Solution Description]
By definition, if $P + Q = O$, then $Q$ is the additive inverse of $P$ (and vice versa). This means $Q = -P$. Therefore, the correct statement is that $Q$ is the additive inverse of $P$.

Key Concept: Additive Inverse of a Matrix

b) $\begin{bmatrix} -3 & 1 \\ -5 & 2 \end{bmatrix}$
[Solution Description]
The additive inverse of matrix $A$ is $-A$, which means we negate every element of $A$.
Given $A = \begin{bmatrix} 3 & -1 \\ 5 & -2 \end{bmatrix}$,
$-A = \begin{bmatrix} -3 & 1 \\ -5 & 2 \end{bmatrix}$
Thus, the additive inverse of $A$ is $\begin{bmatrix} -3 & 1 \\ -5 & 2 \end{bmatrix}$.

Key Concept: Unique Additive Inverse

c) The additive inverse of a matrix is unique.
[Solution Description]
According to the properties of additive inverses in matrices:
1. The additive inverse of a matrix $A$ is unique and is given by $-A$.
2. Adding a matrix to its additive inverse results in the null matrix: $A + (-A) = O$.
Among the options, the correct statement is that the additive inverse of a matrix is unique.

Key Concept: Additive Inverse Property

b) $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
[Solution Description]
By definition, the sum of a matrix and its additive inverse is the null matrix (zero matrix).
Given:
$B = \begin{bmatrix} 0 & 4 \\ -7 & 6 \end{bmatrix}$
The additive inverse $-B$ is:
$-B = \begin{bmatrix} 0 & -4 \\ 7 & -6 \end{bmatrix}$
Now, adding $B$ and $-B$:
$B + (-B) = \begin{bmatrix} 0+0 & 4-4 \\ -7+7 & 6-6 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
Thus, $B + (-B) = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$.

Key Concept: Additive Inverse of a Matrix

a) $\begin{bmatrix} -5 & 1 \\ -2 & 3 \end{bmatrix}$
[Solution Description]
To find the additive inverse of matrix $A$, we take the negative of each element of $A$.
Given:
$A = \begin{bmatrix} 5 & -1 \\ 2 & -3 \end{bmatrix}$
The additive inverse $-A$ is calculated as:
$-A = \begin{bmatrix} -5 & 1 \\ -2 & 3 \end{bmatrix}$
Therefore, the additive inverse of $A$ is $\begin{bmatrix} -5 & 1 \\ -2 & 3 \end{bmatrix}$.

Key Concept: Additive Inverse, Null Matrix

d) $\begin{bmatrix} 5 & -7 \\ 0 & -3 \end{bmatrix}$
[Solution Description]
To obtain the null matrix $O$ from $X$, we need to add the additive inverse of $X$, denoted as $-X$.
First, find $-X$:
$-X = -\begin{bmatrix} -5 & 7 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} 5 & -7 \\ 0 & -3 \end{bmatrix}$
Now, verify $X + (-X) = O$:
$\begin{bmatrix} -5 & 7 \\ 0 & 3 \end{bmatrix} + \begin{bmatrix} 5 & -7 \\ 0 & -3 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = O$

Key Concept: Matrix Order Consistency, Additive Identity

b) $2 \times 2$
[Solution Description]
The equation $(B + C) + D = B + (C + D)$ holds for any matrix $D$ of the same order as $B$ and $C$ due to the associative property of matrix addition.
Since $B$ is a $2 \times 2$ matrix and $C$ is a $2 \times 2$ null matrix, $D$ must also be a $2 \times 2$ matrix.
This ensures consistency in matrix addition and preserves the additive identity property.

Key Concept: Additive Identity Properties, Matrix Addition

c) $A + O = A$
[Solution Description]
The additive identity property states that for any matrix $A$ and null matrix $O$ of the same order, $A + O = O + A = A$.
For a $2 \times 3$ matrix $A$, the null matrix $O$ must also be $2 \times 3$.
Thus, $A + O = A$ holds true as per the definition.

Key Concept: Additive Identity Property

c) $C + O = O + C = C$
[Solution Description]
The additive identity property states that adding the null matrix $O$ to any matrix $C$ of the same order leaves $C$ unchanged.
Mathematically, this is represented as:
$C + O = O + C = C.$