Matrix Product with Adjugate Matrix

Theorem
Let $\mathbf A = \begin{bmatrix} a \end{bmatrix}_n$ be an invertible square matrix of order $n$.

Let $\mathbf B = \begin{bmatrix} b \end{bmatrix}_n = \mathbf A^{-1}$ be the inverse of $\mathbf A$.

Then $\mathbf B$ is defined as:
 * $b_{ij} = \dfrac {A_{ji}} {\det \mathbf A}$

where:
 * $\det \mathbf A$ is the determinant of $\mathbf A$
 * $A_{ji}$ is the cofactor of $a_{ji} \in \mathbf A$.

Corollary
A square matrix is invertible iff its determinant is not zero.