\begin{definition}[Definition:Determinant/Matrix/Definition 1]
Let $\mathbf A = \sqbrk a_n$ be a square matrix of order $n$.
That is, let:
:$\mathbf A = \begin {bmatrix}
a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\
a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\
 \vdots &  \vdots & \ddots &  \vdots \\
a_{n 1} & a_{n 2} & \cdots & a_{n n} \\
\end {bmatrix}$
Let $\lambda: \N_{> 0} \to \N_{> 0}$ be a permutation on $\N_{>0}$.
The '''determinant''' of $\mathbf A$ is defined as:
:$\ds \map \det {\mathbf A} := \sum_{\lambda} \paren {\map \sgn \lambda \prod_{k \mathop = 1}^n a_{k \map \lambda k} } = \sum_\lambda \map \sgn \lambda a_{1 \map \lambda 1} a_{2 \map \lambda 2} \cdots a_{n \map \lambda n}$
where:
:the summation $\ds \sum_\lambda$ goes over all the $n!$ permutations of $\set {1, 2, \ldots, n}$
:$\map \sgn \lambda$ is the sign of the permutation $\lambda$.
\end{definition}