# Show that if A is both diagonalizable and invertible, then so is A inverse.

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### 1 Answer

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.