GATE Mechanical 2025 | Question: 1

Question

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

• A. $\mathbf{A}^{\mathrm{T}}=\mathrm{A}^{-1}$
• B. $\mathbf{A B}=\mathbf{B} \mathbf{A}$
• C. $(\mathbf{A B})^{\mathrm{T}}=\mathbf{B}^{\mathrm{T}} \mathbf{A}^{\mathrm{T}}$
• D. $\mathbf{A}=\mathbf{A}^{-1}$

Answer 1

For any two conformable matrices $A$ and $B$, the transpose of their product satisfies the reversal law: $(AB)^T = B^T A^T$.

The identity $(AB)^T = B^T A^T$ is a fundamental property of matrix algebra that holds true for all matrices, regardless of whether they are symmetric, skew-symmetric, or identity matrices.

•     Option A and D describe orthogonal and involuntary matrices, respectively, which are not guaranteed for all symmetric matrices.
•   Option B ($AB = BA$) only holds if $A$ and $B$ commute, which is not a general property for symmetric matrices.

$(AB)^T = B^T A^T$ (Option C)