If A is a skew-symmetric matrix and \(n \in N\) such that \((A^n)^T=\lambda A^n\), then \(\lambda\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \((-1)^n\)
For a skew-symmetric matrix, \(A^T = -A\). The transpose of a power is the power of the transpose, since \((AB)^T = B^TA^T\):
\((A^n)^T = (A^T)^n\)
\(= (-A)^n\)
\(= (-1)^nA^n\)
So \(\lambda = (-1)^n\): \(A^n\) is symmetric for even n and skew-symmetric for odd n.