Prove that the matrix $I_n - A$ is invertible Let $A$ be an $n \times n$ matrix with the property that $A^3 = O_n$, where $O_n$ denotes the $n \times n$ matrix which
has all the entries equal to $0$. Let $I_n$ be the $n \times n$ identity matrix. Prove that the matrix $I_n - A$
is invertible, and indicate how you would obtain its inverse.

I assume that you mean you are looking for the inverse of $I_n - A$.

The answer is $I_n+A+A^2.$

as results from the development of 
$$(I_n - A)(I_n+A+A^2)=I_n$$
due to the property $A^3=0_n$.
