Question

Direction: Questions number 19 and 20 are Assertion and Reason based questions. Two statements are given, one labelled Assertion (A) and the other labelled Reason (R). Select the correct answer from the codes (A), (B), (C) and (D) as given below.
Assertion (A) : If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA.
Reason (R) : Product of two diagonal matrices of same order is commutative.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

Matrix multiplication is fundamentally an operation on linear transformations where composition order matters. While specific subsets of matrices commute (like diagonal matrices, scalar matrices, or powers of the same matrix $A^k$), the general algebraic structure of matrices is strictly non-commutative.

Answer 1

The Correct Option is B

Concept: To analyze an Assertion-Reason question involving matrix algebra, we must independently verify the properties of general matrix multiplication and the special case of diagonal matrix multiplication.
• General Matrix Multiplication: Matrix multiplication is non-commutative in general. For two square matrices $A$ and $B$ of the same order $n \times n$, both products $AB$ and $BA$ are always defined and yield matrices of order $n \times n$, but the corresponding elements of $AB$ and $BA$ are typically not equal.
• Diagonal Matrices Property: A diagonal matrix is a square matrix where all non-diagonal elements are zero. The multiplication of two diagonal matrices of the identical order is always commutative because it reduces to the scalar multiplication of their respective corresponding diagonal elements.

Step 1: Verification and analysis of Assertion (A).
Let $A$ and $B$ be two square matrices of the same order $2 \times 2$. Let us consider a specific counterexample to verify the assertion that $AB = BA$ is not always true: \[ A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} \] Now, let us calculate the product matrix $AB$ through row-by-column multiplication: \[ AB = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} = \begin{pmatrix} (1)(2) + (2)(1) & (1)(0) + (2)(2) \\ (3)(2) + (4)(1) & (3)(0) + (4)(2) \end{pmatrix} = \begin{pmatrix} 2+2 & 0+4\\ 6+4 & 0+8 \end{pmatrix} = \begin{pmatrix} 4 & 4 \\ 10 & 8 \end{pmatrix} \] Next, let us calculate the reverse product matrix $BA$ using the same rules: \[ BA = \begin{pmatrix} 2 & 0 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} (2)(1) + (0)(3) & (2)(2) + (0)(4) \\ (1)(1) + (2)(3) & (1)(2) + (2)(4) \end{pmatrix} = \begin{pmatrix} 2+0 & 4+0 \\ 1+6 & 2+8 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ 7 & 10 \end{pmatrix} \] Comparing the results element-wise: \[ \begin{pmatrix} 4 & 4 \\ 10 & 8 \end{pmatrix} \neq \begin{pmatrix} 2 & 4 \\ 7 & 10 \end{pmatrix} \quad \implies \quad AB \neq BA \] Since we have explicitly demonstrated a case where $AB \neq BA$, it is established that it is indeed not necessary that $AB = BA$ for any two square matrices. Therefore, Assertion (A) is completely TRUE.

Step 2: Verification of Reason (R) and determination of the logical relationship.
Let $C$ and $D$ be two diagonal matrices of order $n \times n$. Let their diagonal representations be: \[ C = \text{diag}(c_1, c_2, \dots, c_n), \quad D = \text{diag}(d_1, d_2, \dots, d_n) \] When we compute the product $CD$, the resulting matrix is also a diagonal matrix whose elements are the products of the corresponding diagonal entries: \[ CD = \text{diag}(c_1 d_1, c_2 d_2, \dots, c_n d_n) \] Similarly, computing the product $DC$ yields: \[ DC = \text{diag}(d_1 c_1, d_2 c_2, \dots, d_n c_n) \] Since $c_i$ and $d_i$ are ordinary scalar numbers (real numbers), their multiplication satisfies the scalar commutative law: \[ c_i d_i = d_i c_i \quad \forall \ i \in \{1, 2, \dots, n\} \] Consequently, we have: \[ CD = DC \] Thus, the product of two diagonal matrices of the same order is always commutative. Therefore, Reason (R) is completely TRUE. Now we evaluate the link: The Reason talks about a very specific subtype of matrices (diagonal matrices) where multiplication happens to be commutative. It does not provide the underlying rationale or explanation as to why general square matrices fail to commute. The non-commutativity of general matrices stems from the row-by-column cross-multiplication behavior, which is unrelated to the special properties of diagonal matrices. Therefore, Reason (R) is not the correct explanation of Assertion (A).