Editing Determinant (section)

=== Decomposition methods ===
Some methods compute <math>\det(A)</math> by writing the matrix as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include the [[LU decomposition]], the [[QR decomposition]] or the [[Cholesky decomposition]] (for [[Positive definite matrix|positive definite matrices]]). These methods are of order <math>\operatorname O(n^3)</math>, which is a significant improvement over <math>\operatorname O (n!)</math>.<ref>{{cite arXiv|last=Camarero|first=Cristóbal|date=2018-12-05|title=Simple, Fast and Practicable Algorithms for Cholesky, LU and QR Decomposition Using Fast Rectangular Matrix Multiplication|class=cs.NA|eprint=1812.02056}}</ref>

For example, LU decomposition expresses <math>A</math> as a product
:<math> A = PLU. </math>
of a [[permutation matrix]] <math>P</math> (which has exactly a single <math>1</math> in each column, and otherwise zeros), a lower triangular matrix <math>L</math> and an upper triangular matrix <math>U</math>.
The determinants of the two triangular matrices <math>L</math> and <math>U</math> can be quickly calculated, since they are the products of the respective diagonal entries.  The determinant of <math>P</math> is just the sign <math>\varepsilon</math> of the corresponding permutation (which is <math>+1</math> for an even number of permutations and is <math> -1 </math> for an odd number of permutations). Once such a LU decomposition is known for <math>A</math>, its determinant is readily computed as

:<math> \det(A) = \varepsilon \det(L)\cdot\det(U). </math>