Linear Algebra - If A is invertible then A+I is invertible

Question

Prove the given statement true, or give a counterexample to show that it is false.

If $A$ is an $n*n$ invertible matrix, then $A+I_{n}$ is invertible.

I did some examples and found the statement to be true. But how do i prove it for general case.

Answer 1

The statement

"If $A$ is an $n*n$ invertible matrix, then $A+I_{n}$ is invertible"

is false.

Indeed it is easy to find a couterexample. Let assume $A=-I$ then, since $A^{-1}=A$, $A$ is invertible but $A+I=[0]$.