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.