Determinant/Recursive/Multilinear/Scalar/Exercise

Let ${\displaystyle {}K}$ be a field and ${\displaystyle {}n\in \mathbb {N} _{+}}$. Show that the determinant

${\displaystyle \operatorname {Mat} _{n}(K)=(K^{n})^{n}\longrightarrow K,M\longmapsto \det M,}$

fulfills (for arbitrary ${\displaystyle {}k\in \{1,\ldots ,n\}}$ and arbitrary ${\displaystyle {}n-1}$ vectors ${\displaystyle {}v_{1},\ldots ,v_{k-1},v_{k+1},\ldots ,v_{n}\in K^{n}}$, for ${\displaystyle {}u\in K^{n}}$ and for ${\displaystyle {}s\in K}$) the equality

${\displaystyle {}\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\su\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}=s\det {\begin{pmatrix}v_{1}\\\vdots \\v_{k-1}\\u\\v_{k+1}\\\vdots \\v_{n}\end{pmatrix}}\,.}$