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?

Answer 1

(a) $M=A^4-B^4$
$M^T=(A^4-B^4{)}^T=(A^T{)}^4-(B^T{)}^4$
$=A^4-(-B{)}^4=A^4-B^4=M$
(b) $M=AB-BA$
$M^T=(AB-BA{)}^T=(AB{)}^T-(BA{)}^T$
$=B^T{A}^T-A^T{B}^T$
$=-BA-A(-B)$
$=AB-BA=M$
(c) $M=B^5-A^5$
$M^T=(B^T{)}^5-(A^T{)}^5=-(B^5+A^5)\ne -M$
(d) $M=AB+BA$
$M^T=(AB{)}^T+(BA{)}^T$
$=B^T{A}^T+A^T{B}^T=-BA-AB=-M$