The fact that A is invertible means that all the eigenvalues are non-zero. If A is diagonalizable, then, there exists matrices M and N such that `A = MNM^-1 ` .
Taking the inverse of both sides of this equality gives an expression for `A^-1` .
`A^-1 = (MNM^-1)^-1 = (M^-1)^-1 N^-1 M^-1 = MN^-1 M^-1` . Therefore, the inverse of A is also diagonalizable.