Linear Algebra matrices question.

Question

Let $A,B$ be 2 square matrices of the same size. And the following holds true $AB=A+B$

How do I prove that $(I-B)$ and $(I-A)$ are invertible

Answer 1

Note that if $AB=A+B$ then $-A(I-B)=B$ and $-(I-A)B=A$ and replacing $B$ in the second equation we obtain that $$(I-A)(I-B)=I$$

Answer 2

Hint. Compute $(1-A)(1-B)$ where $1$ is the identity matrix.