If  $A=\left[\begin{array}{cc}{\alpha }^2& 5\\ 5& -\alpha \end{array}\right]$ and  $\det \left(A^{10}\right)=1024$  then  $\alpha =$ 

$$\begin{gathered} \left|A\right|=\left|\begin{array}{cc}{\alpha }^2& 5\\ 5& -\alpha \end{array}\right|=-{\alpha }^3-25 \\ \left|A^{10}\right|={\left|A\right|}^{10}={(-{\alpha }^3-25)}^{10}=1024 \\ \Rightarrow {(-{\alpha }^3-25)}^{10}=2^{10} \\ \therefore -{\alpha }^3-25=2 \\ -{\alpha }^3=27=3^3 \\ {\alpha }^3={(-3)}^3\Rightarrow \alpha =-3 \end{gathered}$$