Chapter 2 - Matrices - 2.3 The Inverse of a Matrix - 2.3 Exercises - Page 73: 65

the statement is correct

let $A$ is invertible , then $(A^{2})^{-1}=(AA)^{-1}=A^{-1}A^{-1}$ let the statement is correct when k=n, then $(A^{n})^{-1}=(A^{-1})^{n}$ Thus $(A^{n+1})^{-1}=(A^{n}A)^{-1}=A^{-1}(A^{n})^{-1}=A^{-1}(A^{-1})^{n}=(A^{-1})^{n+1}$ Therefore $(A^{k})^{-1}=(A^{-1})^{k}$ , k is a positive integer