Prove a matrix is invertible 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.
What is the best way to do this?

Given that $A$ satisfies $A^2−4A−7I=0$ (use Cayley- Hamilton theorem). Let $\gamma_1$ and $\gamma_2$ be eigen value of $A$.
$\gamma_1+\gamma_2=4$.
$\gamma_1\gamma_2=-7$. Therefore $\gamma_1$ and $\gamma_2$ are non zero.
Hence $A$ is invertible matrix
