Method of finding inverse of a Matrix using minimal polynomials

Question

Using a piece from my last question I want to show how to find $A^{-1}$ as a polynomial expression in $A$ of degree < $\deg m_A$ where the leading coefficient of the polynomial is $\dfrac{-1}{m_A(0)}$ where $m_A$ is the minimal polynomial of $A$.

Answer 1

If $m_A(x)=x^m+a_{m-1}x^{m-1}+\cdots+a_1x+a_0$ then since $m_A(A)=0$ implies \begin{align*} &A^m+a_{m-1}A^{m-1}+\cdots+a_1A+a_0I=0\\ &\implies a_0I=-(A^m+a_{m-1}A^{m-1}+\cdots+a_1A)\\ &\implies a_0I=A(-A^{m-1}-a_{m-1}A^{m-2}-\cdots-a_1)\\ &\implies I=A\left(-\dfrac{1}{a_0}A^{m-1}-\dfrac{a_{m-1}}{a_0}A^{m-2}-\cdots-\dfrac{a_1}{a_0}\right)\\ &\implies A^{-1}=-\dfrac{1}{a_0}A^{m-1}-\dfrac{a_{m-1}}{a_0}A^{m-2}-\cdots-\dfrac{a_1}{a_0} \end{align*}