Question

Let \(A\) and \(B\) be real symmetric matrices of the same size. Which one of the following options is correct?

• A. \( A^T = A^{-1} \)
• B. \( AB = BA \)
• C. \( (AB)^T = B^T A^T \)
• D. \( A = A^{-1} \)

Hint:

For symmetric matrices, remember that they commute with each other, meaning \( AB = BA \), and their transposes retain the symmetry property \( A^T = A \).

Answer 1

The Correct Option is B

We are given two real symmetric matrices A and B, and we need to determine which statement is correct.

1. Symmetric Matrix Property:
A matrix A is symmetric if \( A^T = A \), i.e., it is equal to its transpose.

2. Commutativity of Symmetric Matrices:
For two symmetric matrices A and B, it is a known property that they commute, i.e., \( AB = BA \). This is because symmetric matrices can be diagonalized by the same orthogonal matrix, and diagonal matrices commute with each other.

3. Let's analyze the options:
- Option (A): \( A^T = A^{-1} \)
    This is not generally true for symmetric matrices. The inverse of a matrix is not necessarily its transpose, except in special cases like orthogonal matrices, which we are not given here.

- Option (B): \( AB = BA \)
    This is the correct property for symmetric matrices. As explained above, symmetric matrices commute with each other.

- Option (C): \( (AB)^T = B^T A^T \)
    This is always true for any two matrices, not just symmetric ones. This is a standard property of matrix transposition: \( (AB)^T = B^T A^T \).

- Option (D): \( A = A^{-1} \)
    This is not necessarily true. A matrix is equal to its inverse only if it is an involutory matrix (i.e., \( A^2 = I \)), which is not stated in the problem.

Thus, the correct answer is (B).