I started off with the normal procedure:
$(A^k)^{-1}=(A^{-1})^k$, let's try with a base case of k=1, this gives: $(A^1)^{-1}=(A^{-1})^1$ which can be simplified to $A^{-1}=(A^{-1})$. So it is true for k=1. Now I assume it is true for k=p and so I get: $(A^p)^{-1}=(A^{-1})^p$, and if it is true for k=p then it is also true for k=p+1: $(A^{p+1})^{-1}+(A^{-1})^p=(A^{-1})^{p+1}$ But this is clearly not true, so I have for sure made an error because if the quantities from the start are equal, LHS can't have an extra factor of $(A^{-1})^p$ So there is a big flaw in my setup.
You want to show that $(A^{p+1})^{-1}=(A^{-1})^{p+1}$. In other words, $(A^{-1})^{p+1}$ is the inverse of $A^{p+1}$, or equivalently, $A^{p+1}(A^{-1})^{p+1}=I$ where $I$ denotes the identity matrix. But $$ A^{p+1}(A^{-1})^{p+1}=A\times A^p\times(A^{-1})^p\times A^{-1}=A\times I\times A^{-1}=I, $$ hence it is true for $k=p+1$.