Finding an expression of the matrix $A^{-1}$ where $A^2−2 A−3 I =0$

Question

Suppose that A is an n×n matrix satisfying $A^2−2 A−3 I =0$ .Give an expression for $A^{−1}$ in terms of A and the identity matrix I.

I have never done this type of matrix question, so it would be extremely helpful if you can give me some hints.

The way I have thought about doing it is as follows:

$$A^2-2A=3I \implies A(A-2)=3I$$

$$A^{-1}A(A-2)=3IA^{-1}$$

$$\frac{1}{3}I(A-2)=A^{-1} I$$

If my working to this point is correct, I have no idea how to remove I from the right hand side of the equation which will help me to complete writing the expression in terms of A.

Answer 1

$A^{-1} I$ is just $A^{-1}$, so your equation actually reads $$A^{-1} = \frac{1}{3}(A-2I).$$

Answer 2

It should be: $$A^{-1} = \frac{1}{3} (A-2I)$$