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) $C=AB-BA$
$C^T=(AB-BA{)}^T=(AB{)}^T-(BA{)}^T$
$=B^T{A}^T-A^T{B}^T=-BA+AB=C$
(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) $C=AB+BA$
$C^T=(AB+BA{)}^T=(AB{)}^T+(BA{)}^T$
$=-BA-AB=-C$
$\therefore$ Option $C$ is not true.