Matrix/Adjugate matrix/Formula with determinant/Fact

Let ${}K$ be a field, and let ${}M$ denote an ${}n\times n$-matrix over ${}K$.

Then

${}(M^{\operatorname {adj} })\cdot M=(\det M)E_{n}=M\cdot (M^{\operatorname {adj} })\,$

If ${}M$ is invertible, then

${}M^{-1}={\frac {1}{\det M}}\cdot M^{\operatorname {adj} }\,.$

Proof, Write another proof