Definition:Determinant/Matrix/Definition 1

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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:

$\displaystyle \map \det {\mathbf A} := \sum_{\lambda} \paren {\map \sgn \lambda \prod_{k \mathop = 1}^n a_{k \map \lambda k} } = \sum_\lambda \map \sgn \lambda a_{1 \map \lambda 1} a_{2 \map \lambda 2} \cdots a_{n \map \lambda n}$

where:

the summation $\displaystyle \sum_\lambda$ goes over all the $n!$ permutations of $\set {1, 2, \ldots, n}$
$\map \sgn \lambda$ is the sign of the permutation $\lambda$.


In Full

When written out in full, the determinant of $\mathbf A$ is denoted:

$\map \det {\mathbf A} = \begin {vmatrix} a_{1 1} & a_{1 2} & \cdots & a_{1 n} \\ a_{2 1} & a_{2 2} & \cdots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \\ \end {vmatrix}$


Order

The order of a determinant is defined as the order of the square matrix on which it is defined.


Also denoted as

The notation $\size {\mathbf A}$ can be used for $\map \det {\mathbf A}$ but this may be prone to ambiguity.

Some sources omit the brackets: $\det \mathbf A$.

Where ambiguity does not result, either style is acceptable on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Also defined as

While a determinant is a number which is associated with a square matrix, the use of the term for the actual array itself is frequently seen.

Thus we can discuss, for example, the elements, columns and rows of a determinant.

So, similarly to square matrix, we can discuss a determinant of order $n$.


Also see

  • Results about determinants can be found here.