linear algebra - Showing that a Matrix is Nonsingular

Let $n$ be a positive integer and let $F$ be a field. Let $A \in M_{n×n}(F )$ be a matrix for which there exists a matrix $ B \in M_{n×n}(F )$ satisfying $I + A + AB = O$. Show that $A$ is nonsingular.

Since $I + A + AB = O$, we can get
$$I+A(I+B)=0$$
$$A[-(I+B)]=I$$

I know it seems $-(I+B)$ is the inverse of $A$, however, I am not sure how to get
$-(I+B)A=I$. I may choose a wrong way to solve this problem. I am really stuck with this question.

Answer

\begin{align*} I_n + A + AB & = O \\
I_n + A(I_n + B) & = O \\
A(I_n + B) & = -I_n.
\end{align*} Take determinant of both sides gives $$\text{det}(A)\text{det}(I_n+B) = (-1)^n,$$ hence $\text{det}(A)$ is non-zero and $A$ is invertible.