Question

Let $A,B,C$ be $3\times 3$ matrices such that $A$ is symmetric and $B$ and $C$ are skew-symmetric.
Consider the statements
(S1) $A^{13}{B}^{26}-B^{26}{A}^{13}$ is symmetric
(S2) $A^{26}{C}^{13}-C^{-13}{A}^{26}$ is symmetric
Then,

• A. Only S 1 is true
• B. Both S1 and S2 are false
• C. Both S1 and S2 are true
• D. Only S2 is true

Answer 1

Answer: (D)

Given, $A^T=A,B^T=-B,C^T=-C$
Let $M=A^{13}{B}^{26}-B^{26}{A}^{13}$
Then, $M^T={\left(A^{13}{B}^{26}-B^{26}{A}^{13}\right)}^T$
$={\left(A^{13}{B}^{26}\right)}^T-{\left(B^{26}{A}^{13}\right)}^T$
$={\left(B^T\right)}^{26}{\left(A^T\right)}^{13}-{\left(A^T\right)}^{13}{\left(B^T\right)}^{26}$
$=B^{26}{A}^{13}-A^{13}{B}^{26}=-M$
Hence, $M$ is skew symmetric
Let, $N=A^{26}{C}^{13}-C^{13}{A}^{26}$
then, $N^T={\left(A^{26}{C}^{13}\right)}^T-{\left(C^{13}{A}^{26}\right)}^T$
$=-(C{)}^{13}(A{)}^{26}+A^{26}{C}^{13}=N$
Hence, $N$ is symmetric.
$\therefore$ Only S2 is true.