Let be a complex cube root of unity, 1 . Let A = bmatrix & 4 0 & ^2 bmatrix and P = 1 5 bmatrix 1 & 2 -2 & 1 bmatrix . If B = P A P^T and C = P^T (B²⁰ - 2B¹⁴ + 3I) P , where I is the identity matrix of order 2 , then the sum of the diagonal elements of C is :

Step-by-step solution

Given P = 1 5 bmatrix 1 & 2 -2 & 1 bmatrix , we can check that P^T P = 1 5 bmatrix 1 & -2 2 & 1 bmatrix bmatrix 1 & 2 -2 & 1 bmatrix = 1 5 bmatrix 5 & 0 0 & 5 bmatrix = I . Thus, P is an orthogonal matrix, which means P^T = P⁻¹ . We are given B = P A P^T . Pre-multiplying by P^T and post-multiplying by P , we get P^T B P = A . For any positive integer k , P^T B^k P = A^k . The matrix C is given by: C = P^T (B²⁰ - 2B¹⁴ + 3I) P C = P^T B²⁰ P - 2 P^T B¹⁴ P + 3 P^T I P C = A²⁰ - 2A¹⁴ + 3I Since A = bmatrix & 4 0 & ^2 b