1. Characteristic Eqn., Eigenvalues & Eigenvectors For each exercise i-iii below, a matrix A is given. a. Find the characteristic equation for A. b. Find the eigenvalues of A. c. Find the eigenvectors corresponding to each eigenvalue. d. Verify the result of part (c) by showing that Axi = ixi. i. [ A= 1 2 7 8 ii. [ [ 0 0 0 2 0 0 0 2 0 0 0 4 5 1 0 A= 0 5 9 5 1 0 iii. 1 0 A= 0 0 2. Solving Linear Homogeneous Systems of DEs. i. Find the general solution x of the following system of equations using Diagonalization: x1' = x2 x2' = x1 ii. Solve the following initial value problem using Diagonalization: x1' = x1 + 2x2 + x3 x2' = x1 - x3 x3' = x1 + x2 + x3 with the initial conditions: x1(0) = 6; x2(0) = -4; x3(0) = 8 For both parts [i] and [ii], you must do the following: a. Rewrite the system in matrix format is x' = Ax; b. Find the eigenvalues and eigenvectors of matrix A. c. Show that matrix A is diagonalizable. d. Find the matrix of eigenvectors X and its inverse X-1. e. Find the diagonal matrix A = X-1AX. f. Use diagonalization to uncouple the system of DEs to y' = Ay. g. Find the general solution from x = Xy. h. If initial conditions are given, find the complete solution. Reminder: Use row-reduction (elimination) by-hand to find the eigenvectors. \f\f\f\f\f\f