Question

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

• A. $AB{A}^T$ is skew symmetric matrix
• B. $A{B}^T+B{A}^T$ is symmetric matrix
• C. $(A+B)(A-B)$ is symmetric
• D. $(A+I)(B-I)$ is skew symmetric

Answer 1

Answer: (A), (B), (C)

$AB{A}^T$ is skew symmetric matrix
$A{B}^T+B{A}^T$ is symmetric matrix
$(A+B)(A-B)$ is symmetric

$A^T=A$ and $B^T=-B$
(A) ${\left(AB{A}^T\right)}^T={A^T}^T{B}^T{A}^T=-\left(AB{A}^T\right)$

Hence, option A is True

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

Hence, option B is True

(C) ${\left(\right(A+B\left)\right(A-B\left)\right)}^T={(A-B)}^T{(A+B)}^T=(A^T-B^T)\left(A^T{+B}^T\right)=\left(\right(A+B\left)\right(A-B\left)\right)$
Hence, option C is True

(D) ${\left(\right(A+I\left)\right(B-I\left)\right)}^T=(B^T-I)(A^T+1)$
$=-(B+I)(A+I)$

Hence, option D is False.