Abstract Algebra Theory and Applications - Computer Science ...

10.1 MATRIX GROUPS 173We are guaranteed that A −1 exists, since det(A) = 2·3−5·1 = 1 is nonzero.Some other facts about determinants will also prove useful in the courseof this chapter. Let A and B be n × n matrices. From linear algebra wehave the following properties of determinants.• The determinant is a homomorphism into the multiplicative group ofreal numbers; that is, det(AB) = (det A)(det B).• If A is an invertible matrix, then det(A −1 ) = 1/ det A.• If we define the transpose of a matrix A = (a ij ) to be A t = (a ji ), thendet(A t ) = det A.• Let T be the linear transformation associated with an n × n matrix A.Then T multiplies volumes by a factor of | det A|. In the case of R 2 ,this means that T multiplies areas by | det A|.Linear maps, matrices, and determinants are covered in any elementarylinear algebra text; however, if you have not had a course in linear algebra,it is a straightforward process to verify these properties directly for 2 × 2matrices, the case with which we are most concerned.The General and Special Linear GroupsThe set of all n × n invertible matrices forms a group called the generallinear group. We will denote this group by GL n (R). The general lineargroup has several important subgroups. The multiplicative properties ofthe determinant imply that the set of matrices with determinant one is asubgroup of the general linear group. Stated another way, suppose thatdet(A) = 1 and det(B) = 1. Then det(AB) = det(A) det(B) = 1 anddet(A −1 ) = 1/ det A = 1. This subgroup is called the special linear groupand is denoted by SL n (R).Example 3. Given a 2 × 2 matrix( a bA =c dthe determinant of A is ad − bc. The group GL 2 (R) consists of those matricesin which ad − bc ≠ 0. The inverse of A is( )A −1 1 d −b=.ad − bc −c a),