Question

If $A$ and $B$ are symmetric matrices, then $ABA$ is

• A. Symmetric
• B. Skew-symmetric
• C. Diagonal
• D. Triangular

Answer 1

The correct option is A Symmetric

Given $A$ and $B$ are symmetric matrices.

$\Rightarrow A^T=A$ and $B^T=B$

Now, take $(ABA{)}^T$

$\Rightarrow (ABA{)}^T=A^T{B}^T{A}^T$

$\Rightarrow (ABA{)}^T=ABA$

Hence, $ABA$ is also a symmetric matrix.