Product of Matrix with Adjugate equals Determinant by Unit Matrix

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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 $n$.


Proof


Sources