Determinant

Let ${}K$ be a
field,
and let ${}M$ denote an
${}n\times n$-matrix
over ${}K$ with entries ${}a_{ij}$. For
${}i\in \{1,\ldots ,n\}$,
let ${}M_{i}$ denote the ${}(n-1)\times (n-1)$-matrix, which arises from ${}M$, when we remove the first column and the ${}i$-th row. Then one defines recursively the * determinant* of ${}M$ by

- ${}\det M={\begin{cases}a_{11}\,,&{\text{ for }}n=1\,,\\\sum _{i=1}^{n}(-1)^{i+1}a_{i1}\det M_{i}&{\text{ for }}n\geq 2\,.\end{cases}}\,$