Editing Determinant (section)

===Leibniz formula===
{{main|Leibniz formula for determinants}}

==== 3 × 3 matrices ====
The ''Leibniz formula'' for the determinant of a {{math|3 × 3}} matrix is the following:
:<math>\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}
 = aei + bfg + cdh - ceg - bdi - afh.\ </math>
In this expression, each term has one factor from each row, all in different columns, arranged in increasing row order. For example, ''bdi'' has ''b'' from the first row second column, ''d'' from the second row first column, and ''i'' from the third row third column. The signs are determined by how many transpositions of factors are necessary to arrange the factors in increasing order of their columns (given that the terms are arranged left-to-right in increasing row order): positive for an even number of transpositions and negative for an odd number. For the example of ''bdi'', the single transposition of ''bd'' to ''db'' gives ''dbi,'' whose three factors are from the first, second and third columns respectively; this is an odd number of transpositions, so the term appears with negative sign.
[[File:Sarrus rule1.svg|thumb|[[Rule of Sarrus]]]]
The [[rule of Sarrus]] is a mnemonic for the expanded form of this determinant: the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the first two columns of the matrix are written beside it as in the illustration. This scheme for calculating the determinant of a {{math|3 × 3}} matrix does not carry over into higher dimensions.

==== ''n'' × ''n'' matrices ====

Generalizing the above to higher dimensions, the determinant of an <math>n \times n</math> matrix is an expression involving [[permutation]]s and their [[signature (permutation)|signatures]]. A permutation of the set <math>\{1, 2, \dots, n \}</math> is a [[Bijection|bijective function]] <math>\sigma</math> from this set to itself, with values <math>\sigma(1), \sigma(2),\ldots,\sigma(n)</math> exhausting the entire set.  The set of all such permutations, called the [[symmetric group]], is commonly denoted <math>S_n</math>. The signature <math>\sgn(\sigma)</math> of a permutation <math>\sigma</math> is <math>+1,</math> if the permutation can be obtained with an even number of transpositions (exchanges of two entries); otherwise, it is <math>-1.</math>

Given a matrix
:<math>A=\begin{bmatrix}
a_{1,1}\ldots a_{1,n}\\
\vdots\qquad\vdots\\
a_{n,1}\ldots a_{n,n}
\end{bmatrix},</math>
the Leibniz formula for its determinant is, using [[sigma notation]] for the sum,
:<math>\det(A)=\begin{vmatrix}
a_{1,1}\ldots a_{1,n}\\
\vdots\qquad\vdots\\
a_{n,1}\ldots a_{n,n}
\end{vmatrix} = \sum_{\sigma \in S_n}\sgn(\sigma)a_{1,\sigma(1)}\cdots a_{n,\sigma(n)}.</math>

Using [[pi notation]] for the product, this can be shortened into
:<math>\det(A) = \sum_{\sigma \in S_n} \left( \sgn(\sigma) \prod_{i=1}^n a_{i,\sigma(i)}\right)</math>.

The [[Levi-Civita symbol]] <math>\varepsilon_{i_1,\ldots,i_n}</math> is defined on the {{mvar|n}}-[[tuple]]s of integers in <math>\{1,\ldots,n\}</math> as {{math|0}} if two of the integers are equal, and otherwise  as the signature of the permutation defined by the ''n-''tuple of integers. With the Levi-Civita symbol, the Leibniz formula becomes 
:<math>\det(A) = \sum_{i_1,i_2,\ldots,i_n} \varepsilon_{i_1\cdots i_n} a_{1,i_1} \!\cdots a_{n,i_n},</math>
where the sum is taken over all {{mvar|n}}-tuples of integers in <math>\{1,\ldots,n\}.</math>
<ref>{{cite book |last1=McConnell |title=Applications of Tensor Analysis |url=https://archive.org/details/applicationoften0000mcco |url-access=registration |date=1957 |publisher=Dover Publications |pages=[https://archive.org/details/applicationoften0000mcco/page/10 10–17]}}</ref><ref>{{harvnb|Harris|2014|loc=§4.7}}</ref>