Definition:Determinant/Matrix

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}$

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.

Determinant of Order 1
This is the trivial case:

Determinant of Order 3
Let:

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