Let $k$ be a field, let $n\in\mathbb N$ with $n\ge 1$, let $M_n(k)$ be the ring of $n\times n$ matrices over $k$, let $I_n\in M_n(k)$ denote the identity matrix, and let $A\in M_n(k)$ be invertible. Let $\det: M_n(k)\to k$ denote the determinant map, and let $t$ be an indeterminate over $k$. Let $\chi_A\in k[t]$ be the [characteristic polynomial](/page/Characteristic%20Polynomial) of $A$, defined by $\chi_A(t)=\det(tI_n-A)$. Write

\begin{align*} \chi_A(t)=t^n+c_{n-1}t^{n-1}+\cdots+c_1t+c_0 \end{align*}

with coefficients $c_0,c_1,\dots,c_{n-1}\in k$. Define $c_n:=1$ and use the convention $A^0=I_n$. Then $c_0\ne 0$, and

\begin{align*} A^{-1}=-c_0^{-1}\sum_{j=1}^n c_j A^{j-1}. \end{align*}

Equivalently,

\begin{align*} A^{-1}=-c_0^{-1}(A^{n-1}+c_{n-1}A^{n-2}+\cdots+c_2A+c_1I_n). \end{align*}