Properties of inverse Matrices

Question

Is it possible for $A^3$ to be an Identity matrix without A being invertibe?

If I do the following would it be correct:

$A$.$A^2$= I

Therefore A has to be invertible

Answer 1

Yes, if $A^3=I$ then $\det A^3 =\det I$

We know that $\det I=1$ hence $\det A^3=1$ which means $\det A \neq 0$ . Therefore $A$ is invertible.

Answer 2

You are fully correct: from $A^3=I$ you get $$ AA^2=I=A^2A $$ so $A^2$ qualifies as the inverse of $A$.