Determinant/Recursive/Multilinear/Scalar/Exercise

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

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

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

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

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