Let $M$ be an invertible $n \times n$ matrix whose minimal polynomial $q_M$ and characteristic polynomial are equal. Suppose as well that the $M$ has entries in an algebraically closed field.
How do we show that the minimal polynomial of $M^{-1}$ is given by $q_M(0)^{-1} x^n q_M(\frac{1}{x}) $? ($x$ is the indeterminate in the polynomial ring over the base field $k$?