Editing Linear algebra (section)

===Determinant===
{{main|Determinant}}
The ''determinant'' of a square matrix {{mvar|A}} is defined to be<ref>{{Harvard citation text|Katznelson|Katznelson|2008}} pp. 76&ndash;77, § 4.4.1&ndash;4.4.6</ref>
:<math>\sum_{\sigma \in S_n} (-1)^{\sigma} a_{1\sigma(1)} \cdots a_{n\sigma(n)}, </math>
where {{math|''S<sub>n</sub>''}} is the [[symmetric group|group of all permutations]] of {{mvar|n}} elements, {{mvar|σ}} is a permutation, and {{math|(−1)<sup>''σ''</sup>}} the [[parity of a permutation|parity]] of the permutation. A matrix is [[invertible matrix|invertible]] if and only if the determinant is invertible (i.e., nonzero if the scalars belong to a field).

[[Cramer's rule]] is a [[closed-form expression]], in terms of determinants, of the solution of a [[system of linear equations|system of {{mvar|n}} linear equations in {{mvar|n}} unknowns]]. Cramer's rule is useful for reasoning about the solution, but, except for {{math|1=''n'' = 2}} or {{math|3}}, it is rarely used for computing a solution, since [[Gaussian elimination]] is a faster algorithm.

The ''determinant of an endomorphism'' is the determinant of the matrix representing the endomorphism in terms of some ordered basis. This definition makes sense since this determinant is independent of the choice of the basis.