Question

If \(A\) is symmetric, \(P\) and \(B\) are skew symmetric matrices, then which of the following is correct?

• A. \(A^{13}B^{26} - B^{26}A^{13}\) is a symmetric matrix
• B. \(A^{26}P^{13} - P^{13}A^{26}\) is a skew symmetric matrix
• C. \(P^{13}A^{26} + A^{26}P^{13}\) is a skew symmetric matrix
• D. \(P^{13}B^{26} + B^{13}P^{26}\) is a skew symmetric matrix

Hint:

An odd power of a skew-symmetric matrix is skew-symmetric.
An even power of a skew-symmetric matrix becomes symmetric.
Any power of a symmetric matrix remains symmetric.

Answer 1

The Correct Option is C

Concept:
• For a symmetric matrix \(A\), \(A^T = A\). Thus, \( (A^n)^T = (A^T)^n = A^n \).
• For a skew-symmetric matrix \(M\), \(M^T = -M\). Thus, \( (M^n)^T = (-1)^n M^n \).
• A matrix \(X\) is symmetric if \(X^T = X\), and skew-symmetric if \(X^T = -X\).
• Reversal law for transposes: \((XY)^T = Y^T X^T\).

Step 1: Determine the behavior of individual powers of the given matrices
Given \(A^T = A\), \(P^T = -P\), and \(B^T = -B\).
\[ (A^{26})^T = (A^T)^{26} = A^{26} \quad (\text{symmetric}) \] \[ (P^{13})^T = (P^T)^{13} = (-P)^{13} = -P^{13} \quad (\text{skew-symmetric}) \]

Step 2: Test option C for skew-symmetry using the transpose operator
Let \(X = P^{13}A^{26} + A^{26}P^{13}\).
Taking the transpose of \(X\): \[ X^T = (P^{13}A^{26} + A^{26}P^{13})^T \] \[ = (P^{13}A^{26})^T + (A^{26}P^{13})^T \] \[ = (A^{26})^T (P^{13})^T + (P^{13})^T (A^{26})^T \] \[ = A^{26}(-P^{13}) + (-P^{13})A^{26} \] \[ = -A^{26}P^{13} - P^{13}A^{26} \] \[ = -(P^{13}A^{26} + A^{26}P^{13}) = -X \] Since \(X^T = -X\), it is a skew-symmetric matrix.