Question

If $A$ is a symmetric and $B$ is a skew symmetric matrix, then which of the following is correct?

• A. $ABA^{T}$ is a symmetric matrix
• B. $AB^{T}+BA^{T}$ is a symmetric matrix
• C. $\left(A + B\right)\left(A - B\right)$ is a skew symmetric matrix
• D. $\left(A + I\right)\left(B - I\right)$ is a skew symmetric matrix

Answer 1

Answer: (B)

$A^T=A, B^T=-B$
$\left(A B A^T\right)^T=\left(A^T\right)^T B^T A^T=-A B A^T$ (Skew symmetric)
$\left(A B^T+B A^T\right)^T=\left(A B^T\right)^T+\left(B A^T\right)^T$
$=B A^T+A B^T$
$=B A-A B=B A^T+A B^T$ (Symmetric)
$[(A+B)(A-B)]^T=(A-B)^T(A+B)^T$
$=\left(A^T-B^T\right)\left(A^T+B^T\right)=(A+B)(A-B)$ (Symmetric)
$[(A+I)(B-I)]^T=[B-I])^T([A+I]^T$
$=\left[B^T-I^T\right]\left[A^T+I^T\right]$
$=[-B-I][A+I]$ (neither symmetric nor skew symmetric)