Abstract Algebra Theory and Applications - Computer Science ...

2.2 DEFINITIONS AND EXAMPLES 43matrix( ) a bA =c dis in GL 2 (R) if there exists a matrix A −1 such that AA −1 = A −1 A = I,where I is the 2 × 2 identity matrix. For A to have an inverse is equivalentto requiring that the determinant of A be nonzero; that is, det A = ad−bc ≠0. The set of invertible matrices forms a group called the general lineargroup. The identity of the group is the identity matrixThe inverse of A ∈ GL 2 (R) isA −1 =I =1ad − bc( 1 00 1).( d −b−c aThe product of two invertible matrices is again invertible. Matrix multiplicationis associative, satisfying the other group axiom. For matrices it isnot true in general that AB ≠ BA; hence, GL 2 (R) is another example of anonabelian group.Example 8. Let1 =I =J =K =( ) 1 00 1( ) 0 1−1 0( ) 0 ii 0( ) i 0,0 −iwhere i 2 = −1. Then the relations I 2 = J 2 = K 2 = −1, IJ = K, JK = I,KI = J, JI = −K, KJ = −I, and IK = −J hold. The set Q 8 ={±1, ±I, ±J, ±K} is a group called the quaternion group. Notice that Q 8is noncommutative.Example 9. Let C ∗ be the set of nonzero complex numbers. Under theoperation of multiplication C ∗ forms a group. The identity is 1. If z = a+biis a nonzero complex number, thenz −1 = a − bia 2 + b 2).