Let $\mathrm{A}=\left(\begin{array}{cc}1& 2\\ -2& -5\end{array}\right)$. Let $\mathrm{\alpha }, \mathrm{\beta }\in \mathrm{ℝ}$ be such that ${\mathrm{\alpha A}}^2+\mathrm{\beta A}=2\mathrm{I}$. Then $\mathrm{\alpha }+\mathrm{\beta }$ is equal to 

Characteristic equation of matrix A

$\begin{array}{l}|\mathrm{A}-\mathrm{\lambda I}|=0\\ \left|\begin{array}{cc}1-\mathrm{\lambda }& 2\\ -2& -5-\mathrm{\lambda }\end{array}\right|=0\\ \Rightarrow \mathrm{A} \mathrm{satisfy} \mathrm{the} \mathrm{equation} {\mathrm{\lambda }}^2+4\mathrm{\lambda }=1\\ \Rightarrow {\mathrm{A}}^2+4\mathrm{A}=\mathrm{I}\\ \Rightarrow 2{\mathrm{A}}^2+8\mathrm{A}=2\mathrm{I} ....\left(1\right)\end{array}$

Given that     ${\mathrm{\alpha A}}^2+\mathrm{\beta A}=2\mathrm{I} ....\left(2\right)$

Comparing equation (1) & (2) we get

      $\begin{array}{l}\mathrm{\alpha }=2,\mathrm{\beta }=8\\ \therefore \mathrm{\alpha }+\mathrm{\beta }=10\end{array}$

Ans. (D) (10)