Question
If \[A = \left[ \begin{array}{l}
\alpha \,\,\,\,\,\,\,2\\
2\,\,\,\,\,\,\,\,\alpha
\end{array} \right]\] and $$\left| {{A^3}} \right| = 125$$ then the value of $$\alpha $$ is
A.
$$ \pm 1$$
B.
$$ \pm 2$$
C.
$$ \pm 3$$
D.
$$ \pm 5$$
Answer :
$$ \pm 3$$
Solution :
\[A = \left[ \begin{array}{l}
\alpha \,\,\,\,\,\,\,2\\
2\,\,\,\,\,\,\,\,\alpha
\end{array} \right]{\rm{ and }}\left| {{A^3}} \right| = 125\]
$$\eqalign{
& \Rightarrow \,\,{\left| A \right|^3} = 125 \cr
& {\text{Now, }}\left| A \right| = {\alpha ^2} - 4 \cr
& \Rightarrow \,\,{\left( {{\alpha ^2} - 4} \right)^3} = 125 = {5^3} \cr
& \Rightarrow \,\,{\alpha ^2} - 4 = 5 \cr
& \Rightarrow \,\,\alpha = \pm 3 \cr} $$