Let A = ( cases 1 & 2 - 2 & - 5 cases ) . Let α, β ∈ ℝ be such that...

We know that characteristic equation of matrix $A$ is given by,

\left|A - \lambdaI\right| = 0$</p><p>$⇒ \left|$\begin{cases}1 - \lambda & 2 \\ - 2 & - 5 - \lambda\end{cases}$\right| = 0$</p><p>$⇒ {\lambda}^{2} + 4\lambda = 1$</p><p>$⇒ {A}^{2} + 4A = I$</p><p>Multiplying by$2$both side we get,</p><p>$⇒ 2{A}^{2} + 8A = 2I ⋯ \left(1\right)$</p><p>Given that$\alpha{A}^{2} + \betaA = 2I ⋯ \left(2\right)$</p><p>Now on Comparing equation \left(1\right) \text{ and } \left(2\right)$ we get

$\alpha = 2, \beta = 8$

$∴ \alpha + \beta = 10$