If $B$ is invertible and $A+B=AB$, prove that $A$ is also invertible

Question

Let $A,B$ be $n \times n$ matrices. If $B$ is invertible and $A+B=AB$, how to prove that $A$ is also invertible?

Answer 1

Hint: Prove that $A(\operatorname{Id}-B^{-1})=\operatorname{Id}$.

Answer 2

IF $B$ is invertible: $$ A+B=AB \Rightarrow (A+B)B^{-1}=A \Rightarrow AB^{-1}+I=A $$ so: $$ A(I-B^{-1})=I $$

Answer 3

$A(B-E)B^{-1}=E$ implies invertibility of $A$.

Answer 4

$A+B=AB$

$\implies B=AB-A$

$\implies B=A(B-I)$.

This gives $det(B)=det(A).det(B-I)$

$\implies k=det(A)det(B-I)$, where $k\ne 0$. Hence $det(A)\ne 0$.