Showing that $I-A$ is non-singular

Question

If $A \in M_n(R) $ with $A^3=0$, show that $I - A$ is non-singular with $(I-A)^{-1}=I+A+A^2$.

How could I approach this?

Answer 1

Notice that $(I-A)(I + A + A^2) = I + A + A^2 - A - A^2 - A^3 = I - A^3 = I$ since $A^3 = 0$

Answer 2

For any martices $A,B$,

If $\ A B=I$, $A=B^{-1} \& B=A^{-1}$

$\ (I-A)(I+A+A^2)=I^3-A^3$ as $AI=IA=A$

Since $A^3=0$, $(I-A)(I+A+A^2)=I$

Thus, $(I-A)^{-1}=I+A+A^2$.