Editing Square matrix (section)

===Determinant===
{{Main|Determinant}}
[[File:Determinant example.svg|thumb|300px|right|A linear transformation on <math>\mathbb{R}^2</math> given by the indicated matrix. The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the [[orientation (mathematics)|orientation]], since it turns the counterclockwise orientation of the vectors to a clockwise one.]]

The ''determinant'' <math>\det(A)</math> or <math>|A|</math> of a square matrix <math>A</math> is a number encoding certain properties of the matrix. A matrix is invertible [[if and only if]] its determinant is nonzero. Its [[absolute value]] equals the area (in <math>\mathbb{R}^2</math>) or volume (in <math>\mathbb{R}^3</math>) of the image of the unit square (or cube), while its sign corresponds to the orientation of the corresponding linear map: the determinant is positive if and only if the orientation is preserved.

The determinant of 2×2 matrices is given by
<math display="block">\det \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc.</math>
The determinant of 3×3 matrices involves 6 terms ([[rule of Sarrus]]). The more lengthy [[Leibniz formula for determinants|Leibniz formula]] generalizes these two formulae to all dimensions.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition III.2.1 }}</ref>

The determinant of a product of square matrices equals the product of their determinants:<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Theorem III.2.12 }}</ref>
<math display="block">\det(AB) = \det(A) \cdot \det(B)</math>
Adding a multiple of any row to another row, or a multiple of any column to another column, does not change the determinant. Interchanging two rows or two columns affects the determinant by multiplying it by −1.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Corollary III.2.16 }}</ref> Using these operations, any matrix can be transformed to a lower (or upper) triangular matrix, and for such matrices the determinant equals the product of the entries on the main diagonal; this provides a method to calculate the determinant of any matrix. Finally, the [[Laplace expansion]] expresses the determinant in terms of [[minor (linear algebra)|minors]], i.e., determinants of smaller matrices.<ref>{{Harvard citations |last1=Mirsky |year=1990 |nb=yes |loc=Theorem 1.4.1 }}</ref> This expansion can be used for a recursive definition of determinants (taking as starting case the determinant of a 1×1 matrix, which is its unique entry, or even the determinant of a 0×0 matrix, which is 1), that can be seen to be equivalent to the Leibniz formula. Determinants can be used to solve [[linear system]]s using [[Cramer's rule]], where the division of the determinants of two related square matrices equates to the value of each of the system's variables.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Theorem III.3.18 }}</ref>