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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,
Only $S 2$ is true
Only $S 1$ is true
Both $S 1$ and $S 2$ are false
Both $S1$ and $S2$ are true
Solution
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.
