Adjugate Formula for the Matrix Inverse

Let $n$ be a natural number with $n\ge 2$, and let $A$ be an invertible $n\times n$ real matrix with determinant $\det A\ne 0$. Then

where $\operatorname{adj}(A)$ is the adjugate of $A$. Equivalently, for all $i,j\in\{1,\dots,n\}$,

where $C_{ji}(A)$ is the $(j,i)$ cofactor of $A$.

In particular, each entry of $A^{-1}$ is a polynomial in the entries of $A$ divided by $\det A$.