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

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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 determinant 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


Examples

Order $2$ Matrix

$\begin {pmatrix} a & b \\ c & d \end {pmatrix}^{-1} = \dfrac 1 {a d - b c} \begin {pmatrix} d & -b \\ -c & a \end {pmatrix}$


Sources