Quadratic Equation with Matrix [Prove Invertible]

Question

The problem is:

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

The hint given is:

"We are trying to a matrix that is the inverse of A."

I completed the square and was proceeding to take the square root of both sides when I realized the identity matrix has multiple square roots.

Answer 1

Check that $$ AB=BA=I$$ where $B=7^{-1}(A-4I)$.