Matrices & Determinants
Symmetric and skew-symmetric matrices
nta_pyq_2023_jan
Grade None
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,
Only S2 is true
Only S1 is true
Both S1 and S2 are false
Both S1 and S2 are true
Step-by-Step Solution
Key Concept: Use transpose properties: $(AB)^T = B^TA^T$, and powers of symmetric/skew-symmetric matrices retain their symmetry type depending on the parity of the exponent
$A^T = A$, $B^T = -B$, $C^T = -C$. $B^{26}$ is symmetric, $A^{13}$ is symmetric. Let $M = A^{13}B^{26} - B^{26}A^{13}$: $M^T = (B^{26})^T(A^{13})^T - (A^{13})^T(B^{26})^T = B^{26}A^{13} - A^{13}B^{26} = -M$. So S1 is skew-symmetric (false). $C^{13}$ is skew-symmetric, $A^{26}$ is symmetric. Let $N = A^{26}C^{13} - C^{13}A^{26}$: $N^T = (C^{13})^T(A^{26})^T - (A^{26})^T(C^{13})^T = -C^{13}A^{26} + A^{26}C^{13} = N$. So S2 is symmetric (true). Answer: (1)
Correct Answer: Only S2 is true