Product of Matrix with Adjugate equals Determinant by Unit Matrix

Theorem
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.

Let $\map \det {\mathbf A}$ be the determinant of $\mathbf A$.

Let $\adj {\mathbf A}$ be the determinant of $\mathbf A$.

Then:
 * $\paren {\adj {\mathbf A} } \mathbf A = \map \det {\mathbf A} \mathbf I = \mathbf A \paren {\adj {\mathbf A} }$

where $\mathbf I$ denotes the unit matrix of order $n$.