Consider the matrices A = bmatrix 2 & -2 4 & -2 bmatrix and B = bmatrix 3 & 9 1 & 3 bmatrix . If matrices P and Q are such that PA = B and AQ = B , then the absolute value of the sum of the diagonal elements of 2(P + Q) is _______.

Step-by-step solution

Given A = bmatrix 2 & -2 4 & -2 bmatrix , we find its determinant: |A| = (2)(-2) - (-2)(4) = 4 Since |A| 0 , A is invertible. The inverse of A is: A⁻¹ = 1 4 bmatrix -2 & 2 -4 & 2 bmatrix = bmatrix -1/2 & 1/2 -1 & 1/2 bmatrix We are given PA = B P = BA⁻¹ and AQ = B Q = A⁻¹B . Using the cyclic property of trace Tr (XY) = Tr (YX) , we have: Tr (P) = Tr (BA⁻¹) = Tr (A⁻¹B) = Tr (Q) Let us calculate Tr (P) by finding the diagonal elements of P = BA⁻¹ : P = bmatrix 3 & 9 1 & 3 bmatrix bmatrix -1/2 & 1/2 -1 & 1/2 bmatrix T