Let A = bmatrix 4 & 7 0 & 3 bmatrix and P = 1 2 bmatrix 1 & 1 -1 & 1 bmatrix . If B = P A P^T , then the sum of all the elements of the matrix B^5 is :

Step-by-step solution

First, observe that P is an orthogonal matrix because P P^T = 1 2 bmatrix 1 & 1 -1 & 1 bmatrix bmatrix 1 & -1 1 & 1 bmatrix = bmatrix 1 & 0 0 & 1 bmatrix = I . Thus, P^T = P⁻¹ . We are given B = P A P^T . This implies that B^5 = P A^5 P^T . The sum of all elements of a 2 2 matrix M can be written as e^T M e , where e = bmatrix 1 1 bmatrix . So, the sum of all elements of B^5 is: S = e^T B^5 e = e^T (P A^5 P^T) e = (P^T e)^T A^5 (P^T e) Let us compute P^T e : P^T e = 1 2 bmatrix 1 & -1 1 & 1 bmatrix bmatrix 1 1 bmat