Let S be the set of all matrices A = pmatrix a & b 0 & c pmatrix where a, b, c -5, -4, -3, , 4, 5 . The number of matrices in S such that A^6 = I (where I is the 2 2 identity matrix) is _____.

Step-by-step solution

Let A = pmatrix a & b 0 & c pmatrix . Since A is an upper triangular matrix, its 6^ th power will have the diagonal elements a^6 and c^6 . For A^6 = I , we must have a^6 = 1 and c^6 = 1 . Since a, c are integers, the possible values are a, c 1, -1 . We consider the four possible cases for (a, c) : Case 1: a = 1, c = 1 A = pmatrix 1 & b 0 & 1 pmatrix . By induction, A^k = pmatrix 1 & kb 0 & 1 pmatrix . Thus, A^6 = pmatrix 1 & 6b 0 & 1 pmatrix . For A^6 = I , we need 6b = 0 b = 0 . This gives 1 matrix. Case 2: a = -1