If $A$ is invertible, then ${(A^{-1})}^n$=${(A^{n})}^{-1}$ for $n$ being natural?
So, can I interchange the operations of inverse and power matrix?
This property holds for every natural number $n$?
And, if this property doesnt hold for any invertible matrices, what kind of invertible matrice it holds? I think diagonals invertible matrices is a case.
If $A$ is invertible then $(A^{-1})^n=(A^n)^{-1}$ by the arguments in the comments (a consequence of matrix multiplication being associative).
If $A^n$ is invertible, then we just need to show that this implies $A$ is invertible. Suppose $A$ is not invertible. Then there exists an $x\neq 0$ such that $Ax=0$, so that $(A^n)x=A^{n-1}Ax=0$, which would imply $A^n$ is not invertible, a contradiction. Hence $A$ is invertible so again by the comments the result follows.