1. (a) Let A = [ 0 Find all 2x2 matrices B which satisfy the relation AB = BA. (b) Use row reduction and back-substitution to solve the following system of equa- tions: 1 -3 1 2x+y+z x + 2y + z 2x + z = (d) Find the inverse of the matrix A = det = = (c) Find all conditions on a and 6 for which det 1 2 a a 3a -1 ab + b a 0 2 a 1 1 0 1 1 0 1 1 -2 = 0. 2. Consider the following matrix equation: 2 1 1] a 0 b b0 1 B-W = 2 (a) Use the determinant approach of Example 136 in the workbook to find simple conditions on a and b for the system to have a unique solution, infinitely many solutions, and no solution respectively. Put your answers in the following table: Unique Solution Infinitely Many Solutions No Solutions (b) On a pair of a-b axes, clearly identify the regions, lines or points where each of the three conditions hold. (c) In the case that there is a unique solution, find the solution in terms of a and b. Simplify the expressions for x,y and z as well as you can. 3. (a) Show that the following set of vectors is a linearly independent set: 2 112 O (b) Consider the following system of linear equations: 2x+y 3z + 4w y + 2z+w x+y+2w = 6 = 2 = 3 y + 2z = 1 Explain why you expect this system to have a unique solution (x, y, z, w). (c) Use row-reduction and back-substitution to solve the linear system of part (b).