Editing Determinant (section)

=== Multiplicativity and matrix groups ===
The determinant is a ''multiplicative map'', i.e., for square matrices <math>A</math> and <math>B</math> of equal size, the determinant of a [[matrix product]] equals the product of their determinants:
:<math>\det(AB) = \det (A) \det (B)</math>

This key fact can be proven by observing that, for a fixed matrix <math>B</math>, both sides of the equation are alternating and multilinear as a function depending on the columns of <math>A</math>. Moreover, they both take the value <math>\det B</math> when <math>A</math> is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.<ref>Alternatively, {{harvnb|Bourbaki|1998|loc=§III.8, Proposition 1}} proves this result using the [[functoriality]] of the exterior power.</ref>

A matrix <math>A</math> with entries in a [[field (mathematics)|field]] is [[invertible matrix|invertible]] precisely if its determinant is nonzero. This follows from the multiplicativity of the determinant and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by
:<math>\det\left(A^{-1}\right) = \frac{1}{\det(A)} = [\det(A)]^{-1}</math>.

In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size <math>n</math> over a field <math>K</math>) forms a group known as the [[general linear group]] <math>\operatorname{GL}_n(K)</math> (respectively, a [[subgroup]] called the [[special linear group]] <math>\operatorname{SL}_n(K) \subset \operatorname{GL}_n(K)</math>. More generally, the word "special" indicates the subgroup of another [[matrix group]] of matrices of determinant one. Examples include the [[special orthogonal group]] (which if ''n'' is 2 or 3 consists of all [[rotation matrix|rotation matrices]]), and the [[special unitary group]].

Because the determinant respects multiplication and inverses, it is in fact a [[group homomorphism]] from <math>\operatorname{GL}_n(K)</math> into the multiplicative group <math>K^\times</math> of nonzero elements of <math>K</math>. This homomorphism is surjective and its kernel is  <math>\operatorname{SL}_n(K)</math> (the matrices with determinant one). Hence, by the [[first isomorphism theorem]], this shows that <math>\operatorname{SL}_n(K)</math> is a [[normal subgroup]] of <math>\operatorname{GL}_n(K)</math>, and that the [[quotient group]] <math>\operatorname{GL}_n(K)/\operatorname{SL}_n(K)</math> is isomorphic to <math>K^\times</math>.

The [[Cauchy–Binet formula]] is a generalization of that product formula for ''rectangular'' matrices. This formula can also be recast as a multiplicative formula for [[compound matrix|compound matrices]] whose entries are the determinants of all quadratic submatrices of a given matrix.<ref>{{harvnb|Horn|Johnson|2018|loc=§0.8.7}}</ref><ref>{{harvnb|Kung|Rota|Yan|2009|p=306}}</ref>