if for a Matrix A $ A^3 = 0 $ and A has an Inverse is it safe to assume that A = 0?

Question

I'm sorry if this is an obvious question. I searched and haven't found anyone has asked it before (probably because it's so obvious).

I went over some basic exercises and found one that asks me to find an example where $A$ does not equal $0$ and $A^3 = 0$. There are plenty of example Matrices. But on the other hand, I thought isn't $ A^3 $ just $ AAA $ ? And then if I have $AAA = 0$ and continuously multiply by An inverse on the left or right twice I get $A=0$.

I find this intuitive and though it looks obvious, often my intuition has deceived me. I'd be grateful if someone can confirm if right or explain to me if wrong why I can't multiply by $A$ inverse.

Octave gives me a pseudoinverse for $\begin{pmatrix}0&1\\0&0\end{pmatrix}$

Answer 1

If $A^3 = 0$, then $A$ doesn't have an inverse! For: if $A$ had an inverse, then you could calculate $$ \left(A^{-1}\right)^3 A^3 = A^{-1}A^{-1}A^{-1}AAA = A^{-1}A^{-1}AA = A^{-1}A = I. $$ Yet also, $$ \left(A^{-1}\right)^3 A^3 = \left(A^{-1}\right)^3 0 = 0. $$ But $I \neq 0$, so $A$ does not have an inverse at all.

Answer 2

$\det({A^3})=\left(\det A\right)^3=0$, so $\det(A)=0$ and $A$ has no inverse.