If a matrix $A^2$ is invertible, is $A^3$ invertible?

Question

I know how to find out if a matrix $A^2$ is invertible if $A^3$ is invertible, but how can you find out invertibility if it's the the other way around?

Answer 1

A matrix is invertible if and only if it has a nonzero determinant. Since $\det(A^2) = \det(A)\det(A)$, if $\det(A^2) \neq 0$, then $\det(A) \neq 0$.

Answer 2

We only need the definition and associativity.

By definition since $A^2$ is invertible, there exists a matrix $M$ such that $A^2M=I$, where I is the identity matrix. Now, consider the matrix $AM^2$. Then, by associativity, we have the following. $$A^3(AM^2) = A^2(A^2M)M=A^2M=I$$ Therefore, we know that the $A^3$ is invertible and its inverse is $AM^2$.