Inverse of Matrix is Scalar Product of Adjugate by Reciprocal of Determinant

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

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

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

Then:
 * $\mathbf A^{-1} = \dfrac 1 {\map \det {\mathbf A} } \cdot \adj {\mathbf A}$

where $\mathbf A^{-1}$ denotes the inverse of $\mathbf A$

Proof
Let $\mathbf I_n$ denote the unit matrix of order $n$.