Question

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

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

Hint:

Remember the fundamental properties of matrix operations. The reversal rule for transpose, \( (AB)^T = B^T A^T \), and for inverse, \( (AB)^{-1} = B^{-1} A^{-1} \), are always true for any conforming matrices.

Answer 1

The Correct Option is A

Step 1: Analyze General Matrix Laws. The identity \((AB)^T = B^T A^T\) is known as the reversal law of transposes. It is true for any two matrices where the product is defined.
Step 2: Evaluate Symmetric-Specific Options. - \(AB = BA\): This is only true if the product \(AB\) is also symmetric. Symmetry of \(A\) and \(B\) individually does not guarantee commutativity. - \(A^T = A^{-1}\): This is the definition of an orthogonal matrix. Many symmetric matrices are not orthogonal. - \(A = A^{-1}\): This is an involutory matrix.