If \(A=\begin{bmatrix}-k & 0\\ 0 & -k\end{bmatrix}\), \(k \ne 0\), then the value of m in \(\left(A^T\right)^4 = mA\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(-k^3\)
A is a scalar matrix: \(A = -kI\). It is symmetric, so \(A^T = A\), and
\(\left(A^T\right)^4 = A^4 = (-k)^4 I = k^4 I\)
The right-hand side is
\(mA = m(-kI) = -mk\,I\)
Comparing the two (k ≠ 0):
\(-mk = k^4\)
\(m = -k^3\)