Editing Determinant (section)

==== ''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>