Question

If $A$ and $B$ are two symmetric matrices of the same order, then

• A. $A+B$ is symmetric
• B. $A-B$ is symmetric
• C. $AB$ is symmetric
• D. $BAB$ is symmetric

Answer 1

Answer: (A), (B), (D)

$A+B$ is symmetric
$A-B$ is symmetric
$BAB$ is symmetric

Given $A$ and $B$ are two symmetric matrices.
i.e $A^T=A$ and $B^T=B$
1. $(A+B{)}^T=A^T+B^T=A+B$ $\Rightarrow A+B$ is symmetric.
2. $(A-B{)}^T=A^T-B^T=A-B$ $\Rightarrow A-B$ is symmetric.
3. $(AB{)}^T=B^T{A}^T=BA$ $\Rightarrow AB$ is not symmetric. ($\because AB\ne BA$)
4. $(BAB{)}^T=B^T{A}^T{B}^T=BAB$ $\Rightarrow BAB$ is symmetric.
Hence, options $A,B$ and $D$.