Adjugate Matrix

Definition

$\mathrm{adj}(\mathbf{A})={\mathbf{C}}^{⊺}$ where $\mathbf{C}$ is a Cofactor matrix

Facts

$\mathbf{A}\mathrm{adj}(\mathbf{A})=\mathrm{adj}(\mathbf{A})\mathbf{A}=\det (\mathbf{A})\mathbf{I}$

$\mathrm{adj}(\mathbf{A})=\det (\mathbf{A}){\mathbf{A}}^{-1}$

${\mathbf{A}}^{-1}=\det (\mathbf{A}{)}^{-1}\mathrm{adj}(\mathbf{A})$