Determinant/Laplace expansion/Fact

Expansion theorem for determinant

Let $K$ be a field, and let $M={\left(a_{ij}\right)}_{ij}$ be an $m\times n$-matrix over $K$. For $i,j\in \{1,\dots ,n\}$, let $M_{ij}$ be the matrix which arises from $M$, by leaving out the $i$-th row and the $j$-th column.

Then

(for $n\ge 2$ and for every fixed $i$ and $j$)

$detM=\sum _{i=1}^n(-1{)}^{i+j}{a}_{ij}det{M}_{ij}=\sum _{j=1}^n(-1{)}^{i+j}{a}_{ij}det{M}_{ij}.$

Proof, Write another proof