Definition:Determinant/Matrix

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 $\N_{> 0}$.

Then 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 $\lambda$.

When written out in full, it is denoted by:
 * $\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}$

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.

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

 * Expansion Theorem for Determinants