If \( A=\begin{bmatrix} k & 2\\ 2 & k\end{bmatrix} \) and \( |A^{3}|=125 \), then the value of \( k \) is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \pm 3 \)
Since \( |A^{3}|=|A|^{3} \),
\( \displaystyle |A|^{3}=125\ \Rightarrow\ |A|=5 \)
Expanding the determinant,
\( \displaystyle |A|=k^{2}-4 \)
\( \displaystyle k^{2}-4=5\ \Rightarrow\ k^{2}=9 \)
\( \displaystyle k=\pm 3 \)