Question

Let $A$ and $B$ be any two $3 \times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

• A. $A ^4- B ^4$ is a symmetric matrix
• B. $AB - BA$ is a symmetric matrix
• C. $B ^5- A ^5$ is a skew-symmetric matrix
• D. $AB + BA$ is a skew-symmetric matrix

Answer 1

Answer: (C)

Given that $A ^{ T }=A, B^{ T }=- B$
(A) $C = A ^4- B ^4$
$C^T=\left(A^4-B^4\right)=\left(A^4\right)^T-\left(B^4\right)^T=A^4-B^4=C$
(B) $C = AB - BA$
$ C^T=(A B-B A)^T=(A B)^T-(B A)^T$
$=B^T A^T-A^T B^T=-B A+A B=C$
(C) $C = B ^5- A ^5$
$C^T=\left(B^5-A^5\right)^T=\left(B^5\right)^T-\left(A^5\right)^T=-B^5-A^5$
(D) $C = AB + BA$
$ C^T=(A B+B A)^T=(A B)^T+(B A)^T $
$=-B A-A B=-C$
$\therefore$ Option $C$ is not true.