Question 2 (Eigenvalue method) Question 2. Find the general solution of the homogeneous linear system below using the eigenvalue method. - (2) y

SOLVE CAREFULLY!! Please Write Clearly and Box the final Answer(s) Label them ( C& D)

11) Find the values of such that the equations have non-trivial solutions. What are the eigenvalues and eigenvectors? (6-2) x + 3y = 0 3x - (2+2)y = 0

Use a calculator for: - Finding eigenvalues and eigenvectors. - Matrix multiplication still required to show how you find generalized - Matrix inversion - Row reduction - Integration

Let 1 A = -2 1 Consider the system of equations = A. (a) Find all eigenvalues of A. (b) Show that 7 = 0 is the only equilibrium solution of the system. %3D (c) Determine whether 7 Ở is a node, or a saddle, or a spiral point. (d) Determine the stability of 7 = 0. 2.

(1 point) Suppose A is a 2 x 2 real matrix with an eigenvalue X = 5 + 3i and corresponding eigenvector Determine a fundamental set (i.e., linearly independent set) of solutions for y Enter your solutions below. Use t as the independent variable in your answers. (t) v = (t) = -1+ i Ay, where the fundamental set consists entirely of real solutions.

Use a calculator for: - Finding eigenvalues and eigenvectors. - Matrix multiplication still required to show how you find generalized - Matrix inversion2.

Consider the linear system X1 = and Find the eigenvalues and eigenvectors for the coefficient matrix. A₂ = 0 JI ← A, V₁ 9 V2 = 81 = = -3 6 - - 4 x. help (numbers) help (matrices) help (numbers) help (matrices) For each eigenpair in the previous part, form a solution of a' = Ax. Use t as the independent variable in your answers.

Consider the linear system -6 x' = [161] 2 x. 10 Find the eigenvalues and eigenvectors for the coefficient matrix. λι and = = 12 = , V2 = [8] [8] help (numbers) help (matrices) help (numbers) help (matrices) Book: Section 3.4 of Notes on Diffy Qs

Explain the connection between straight-line solutions, eigenvectors, and the general solution.