Definition:Determinant/Matrix/Definition 1

Definition
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 $\lambda$.

Also see

 * Equivalence of Definitions of Determinant


 * Expansion Theorem for Determinants