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Q.
Let $A= \begin{pmatrix}1 & 2 \\ -2 & -5\end{pmatrix}$. Let $\alpha, \beta \in R$ be such that $\alpha A^2+\beta A=2 I$. Then $\alpha+\beta$ is equal to -
Characteristic equation of matric A
$ | A -\lambda I |=0 $
$ \begin{vmatrix} 1-\lambda & 2 \\-2 & -5-\lambda\end{vmatrix}=0 $
$\Rightarrow \lambda^2+4 \lambda=1 $
$ \Rightarrow A ^2+4 A = I $
$ \Rightarrow 2 A ^2+8 A =2 I ....$(1)
Given that $\alpha A ^2+\beta A =2 I.....$(2)
Comparing equation (1) & (2) we get
$\alpha=2, \beta=8$
$ \therefore \alpha+\beta=10$
Ans. (D) (10)