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)

Only S2 is true

Answer 2

The correct answer is (D) : Only S2 is true
Given, AT=A,BT=−B,CT=−C
Let M=A13B26−B26A13
Then, MT=(A13B26−B26A13)T
=(A13B26)T−(B26A13)T
=(BT)26(AT)13−(AT)13(BT)26
=B26A13−A13B26=−M
Hence, M is skew symmetric
Let, N=A26C13−C13A26
then, NT=(A26C13)T−(C13A26)T
=−(C)13(A)26+A26C13=N
Hence, N is symmetric.
∴ Only S2 is true.