# Finding inverse from matrix equation

1. Oct 26, 2016

### Mr Davis 97

1. The problem statement, all variables and given/known data
Suppose that a square matrix $A$ satisfies $(A - I)^2 = 0$. Find an explicit formula for $A^{-1}$ in terms of $A$

2. Relevant equations

3. The attempt at a solution
From manipulation we find that $A^2 - 2A + I = 0$ and then $A(2I - A) = I$. This shows that if we right-multiply $A$ by $2I - A$, we get the identity matrix. However, to show that $2I - A$ is an inverse, don't we also have to show that we can left-multiply $A$ by $2I - A$ such that $(2I - A)A = I$? I don't see how we can do that...

2. Oct 26, 2016

### Staff: Mentor

I don't think that showing that $(2I - A)A = A(2I - A)$ should be too difficult.

3. Oct 26, 2016

### Staff: Mentor

Yes.
Multiply it.

4. Oct 26, 2016

### Ray Vickson

There is a standard theorem which states that if a square matrix $A$ has a right (left) inverse $B$, then it also has a left (right) inverse, and that is equal to $B$ as well. You should expand your understanding by trying to prove that theorem.