If $A=\begin{bmatrix}-k&0\\0&-k\end{bmatrix},k≠0$, then the value of m in $(A^T)^4 = mA$ is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $-k^3$
$A^T=\begin{bmatrix}-k&0\\0&-k\end{bmatrix}$
$(A^n)=\begin{bmatrix}(-k)^n&0\\0&(-k)^n\end{bmatrix}$
$(A^T)^4=\begin{bmatrix}(-k)^4&0\\0&(-k)^4\end{bmatrix}$ $(∵A^T=A)$
$=\begin{bmatrix}k^4&0\\0&k^4\end{bmatrix}$
$⇒m\begin{bmatrix}-k&0\\0&-k\end{bmatrix}$
$∴m=-k^3$