Let A=pmatrix1&2-2&-5pmatrix.

Question

Let $A=\begin{pmatrix}1&2\-2&-5\end{pmatrix}$. Let $\alpha,\beta\in\mathbb{R}$ be such that $\alpha A^2+\beta A=2I$. Then $\alpha+\beta$ is equal to

Accepted Answer

$A^2=\begin{pmatrix}-3&-8\8&21\end{pmatrix}$.

$\alpha A^2+\beta A=\begin{pmatrix}-3\alpha+\beta & -8\alpha+2\beta\8\alpha-2\beta & 21\alpha-5\beta\end{pmatrix}=\begin{pmatrix}2&0\0&2\end{pmatrix}$.

From $8\alpha=2\beta$ and $-3\alpha+\beta=2$: $\beta=4\alpha$, so $-3\alpha+4\alpha=2\Rightarrow\alpha=2,\ \beta=8$.

$\alpha+\beta=10$.

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