Definition:Determinant/Matrix

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Definition

Let $\mathbf A = \left[{a}\right]_n$ be a square matrix of order $n$.

That is, let:

$\mathbf A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{bmatrix}$


Let $\lambda: \N_{> 0} \to \N_{> 0}$ be a permutation on $\N_{> 0}$.


Then the determinant of $\mathbf A$ is defined as:

$\displaystyle \det \left({\mathbf A}\right) := \sum_{\lambda} \left({\operatorname{sgn} \left({\lambda}\right) \prod_{k \mathop = 1}^n a_{k \lambda \left({k}\right)}}\right) = \sum_{\lambda} \operatorname{sgn} \left({\lambda}\right) a_{1 \lambda \left({1}\right)} a_{2 \lambda \left({2}\right)} \cdots a_{n \lambda \left({n}\right)}$

where:

the summation $\displaystyle \sum_\lambda$ goes over all the $n!$ permutations of $\left\{{1, 2, \ldots, n}\right\}$.
$\operatorname{sgn} \left({\lambda}\right)$ is the sign of the permutation $\lambda$.


When written out in full, it is denoted by:

$\det \left({\mathbf A}\right) = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{vmatrix}$


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 $\left|{\mathbf A}\right|$ can be used for $\det \left({\mathbf A}\right)$ 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 $\mathsf{Pr} \infty \mathsf{fWiki}$.


Examples

Determinant of Order 1

This is the trivial case:


$\begin{vmatrix} a_{11} \end{vmatrix} = \map \sgn 1 a_{1 1} = a_{1 1}$


Thus the determinant of a single number is that number itself.


Determinant of Order 2

$\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = \map \sgn {1, 2} a_{1 1} a_{2 2} + \map \sgn {2, 1} a_{1 2} a_{2 1} = a_{1 1} a_{2 2} - a_{1 2} a_{2 1}$


Determinant of Order 3

Let:

$\map \det {\mathbf A} = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$


Then:

\(\displaystyle \map \det {\mathbf A}\) \(=\) \(\displaystyle a_{1 1} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{1 2} \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{1 3} \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}\)
\(\displaystyle \) \(=\) \(\displaystyle \map \sgn {1, 2, 3} a_{1 1} a_{2 2} a_{3 3}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \map \sgn {1, 3, 2} a_{1 1} a_{2 3} a_{3 2}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \map \sgn {2, 1, 3} a_{1 2} a_{2 1} a_{3 3}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \map \sgn {2, 3, 1} a_{1 2} a_{2 3} a_{3 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \map \sgn {3, 1, 2} a_{1 3} a_{2 1} a_{3 2}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \map \sgn {3, 2, 1} a_{1 3} a_{2 2} a_{3 1}\)
\(\displaystyle \) \(=\) \(\displaystyle a_{1 1} a_{2 2} a_{3 3}\)
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle a_{1 1} a_{2 3} a_{3 2}\)
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle a_{1 2} a_{2 1} a_{3 3}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle a_{1 2} a_{2 3} a_{3 1}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle a_{1 3} a_{2 1} a_{3 2}\)
\(\displaystyle \) \(\) \(\, \displaystyle - \, \) \(\displaystyle a_{1 3} a_{2 2} a_{3 1}\)

and thence in a single expression as:

$\displaystyle \map \det {\mathbf A} = \frac 1 6 \sum_{i \mathop = 1}^3 \sum_{j \mathop = 1}^3 \sum_{k \mathop = 1}^3 \sum_{r \mathop = 1}^3 \sum_{s \mathop = 1}^3 \sum_{t \mathop = 1}^3 \map \sgn {i, j, k} \map \sgn {r, s, t} a_{i r} a_{j s} a_{k t}$

where $\map \sgn {i, j, k}$ is the sign of the permutation $\tuple {i, j, k}$ of the set $\set {1, 2, 3}$.


The values of the various instances of $\map \sgn {\lambda_1, \lambda_2, \lambda_3}$ are obtained by applications of Parity of K-Cycle.


Note

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 matrices, we can discuss a determinant of order $n$.


Also see

  • Results about determinants can be found here.


Historical Note

The theory of determinants was advanced significantly by Augustin Louis Cauchy.


Sources