Showing that a Matrix is Nonsingular

Question

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.

For a square matrix existence of the right inverse implies existence of the left inverse. — Artem, Dec 21 '16 at 05:31
@Artem Thanks, http://math.stackexchange.com/questions/470159/matrices-left-inverse-is-also-right-inverse — Parisina, Dec 21 '16 at 05:41
Take determinants, you have $|A| |-I = B | = 1$ so $|A| \neq 0$ and $A$ is invertible. — Arin Chaudhuri, Dec 21 '16 at 05:41

score 1 · Accepted Answer · edited Dec 24 '16 at 17:42

1

\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.

edited Dec 24 '16 at 17:42

Parisina

1,856

answered Dec 21 '16 at 09:39

Showing that a Matrix is Nonsingular

1 Answers1