Question

If $A=\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]$ and |A³| = 125, then α = ___________.

Answer 1

Given:
$A=\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]$
|A³| = 125



$$\begin{gathered} \mathrm{Now}, \\ \left|A^3\right|=125 \\ \Rightarrow {\left|A\right|}^3=5^3\left(\because \left|A^n\right|={\left|A\right|}^n\right) \\ \Rightarrow \left|A\right|=5 \\ \Rightarrow \left|\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right|=5 \\ \Rightarrow {\alpha }^2-4=5 \\ \Rightarrow {\alpha }^2=5+4 \\ \Rightarrow {\alpha }^2=9 \\ \Rightarrow \alpha =\pm 3 \end{gathered}$$

Hence, α = ±3.