Question

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

• A. $AB{A}^T$ is a symmetric matrix
• B. $A{B}^T+B{A}^T$ 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)

$A^T=A,B^T=-B$
${\left(AB{A}^T\right)}^T={\left(A^T\right)}^T{B}^T{A}^T=-AB{A}^T$ (Skew symmetric)
${\left(A{B}^T+B{A}^T\right)}^T={\left(A{B}^T\right)}^T+{\left(B{A}^T\right)}^T$
$=B{A}^T+A{B}^T$
$=BA-AB=B{A}^T+A{B}^T$ (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)