a) Let A be a real symmetric matrix, and let x₁ and x2 be eigenvectors belonging to real eigenvalues A1 and A2, respectively. By considering the product xAx₁, show that when A1 and A2 are distinct, the eigenvectors X1 and X2 are orthogonal. b) Show that x₁ = 1 2 -().- (F)- and x2 = are eigenvectors of the matrix A 21 and determine the corresponding eigenvalues. 2 1 c) Using the vectors in the preceding part, or otherwise, find the third eigenvalue of A and a corresponding eigenvector. d) Determine an orthogonal matrix P such that Pt AP is diagonal.