For each of the following ODEs determine if the eigenvalues are (a) real and distinct, (b) repeated, (c) complex conjugate pairs: 1. +2x+3x = 0 2. 3. 4. + 4x + x = 0 + 4x + 4x = 0 + 3x = 0

When the eigenvalues of [a b] d A = are 1₁ 3 and 12₂ a = 0 and d = -3 a = -3 and d = 0 a = 3 and d = 0 = = a = 0 and d = 3 a = 0 and d = 0 3 and d = 3 a = 0, what are the possible values of a and d? (Select all that apply.)

The matrix (image attached) has three distinct real eigenvalues if and only if   Fill in blank < k < Fill in blank

x1 = x2 – 4x + 4x3 %3D 4 x½ = -3x2 - V3x, + V3x1 +X3 x3 = 7x2 - 2x1 Find the eigenvalues and eigenvectors by using the above equations.