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The $2 \times 2$ matrix ${A}$ satisfies ${A}^2 - 4 {A} - 7 {I} = {0}$ where ${I}$ is the $2 \times 2$ identity matrix. Prove that ${A}$ is invertible.

I have tried to solve it like a quadratic, but that doesn't work. Any help is appreciated!

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$A^2-4A-7I=0$ $\Longrightarrow$ $A^2-4A=7I$ $\Longrightarrow$ $A\cdot \frac17(A-4I)=I$. So $A$ is invertible. Its inverse is $\frac17(A-4I)$.

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