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 ${}i,j\in \{1,\ldots ,n\}$, let ${}M_{ij}$ be the matrix which arises from ${}M$, by leaving out the ${}i$-th row and the ${}j$-th column.

Then

(for ${}n\geq 2$ and for every fixed ${}i$ and ${}j$)

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

Proof, Write another proof