Question

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Let \(A\) and \(B\) be two symmetric matrices of order 3. Assertion (A): \((AB)\) and \((BA)\) are symmetric matrices.
Reason (R): \(AB\) is a symmetric matrix, if matrix multiplication of \(A\) with \(B\) is commutative. In the light of the above statements, choose the most appropriate answer from the options given below:

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

Matrix multiplication is not commutative in general. Always check \(AB = BA\) before assuming symmetry.

Answer 1

The Correct Option is D

Concept:
• If \(A\) is symmetric, then \(A^T = A\)
• For product: \[ (AB)^T = B^T A^T \]

Step 1: Check assertion. \[ (AB)^T = B^T A^T = BA \] So: \[ AB \text{ is symmetric if } AB = BA \] Thus:
• \(AB\) is not always symmetric
• \(BA\) also not necessarily symmetric Hence Assertion is false.

Step 2: Check reason. If: \[ AB = BA \] Then: \[ (AB)^T = BA = AB \] So \(AB\) is symmetric. Thus Reason is true.

Step 3: Final conclusion. \[ \boxed{(4)\ (A)\ \text{is not correct but}\ (R)\ \text{is correct}} \]