Problem 1.Maximize x_1 x_2subject to x_1 + 3x_2 0 3x_1 + 2x_2 3x1, x2 0.A. Sketch the feasible region in the (x1, x2) space.B. Identify the regions in the (x1, x2) space where the slack variables (say x3 and x4) are equal to zero.C. Solve the problem geometrically/graphically. Problem 4. Does the following matrix have an inverse? If the answer is yes, find A1: Problem 5. Use the Guass-Jordan method to determine whether each of the following linear systems has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions (if any exist) (i) x_1 + x_2 + x_4 = 3x_2 + x_3 = 4x_1 + 2x_2 + x_3 + x_4 = 8 (ii) x_1 + x_2 = 1 2x_1 + x_2 = 33x_1 + 2x_2 = 4 (iii) 2x_2 + 2x_3 = 4x_1 + 2x_2 x_3 = 4x_2 x_3 = 0(iv) x_1 + x_2 = 2x_2 + 2x_3 = 3x_2 + x_3 = 31 Problem 6. Determine the basic and nonbasic variables for all four system of equa- tions in Problem 5. [Note that after Guass-Jordan method has been applied to any linear system, a variable that appears with a coefficient of 1 in a single equation and a coefficient of 0 in all other equations is called a basic variable.]

Problem 7 (30 points) Determine whether the following collection of vectors are linear indepen dent or dependent? Provide details to support your claim. a. -Oo -N. a3 = b. a1 NW- MON 12 C. O - O WN- WNE a3 a = d. a3 -N e. a2 S NA 0 - Problem 2 (10 points) Find the determinants of the following matrices: (i) A = 0 5 -2 1 (ii) A = 2 4 -2 A 3 1 0 1 Problem 3 (10 points) Find the rank of the following matrices: a. A = 2 2 -1 0 1 2 1 -2 -3 1 -1 b. A= -2 2 -1 2 1 2 2 5 Problem 4 (5 points) Does the following matrix have an inverse? If the answer is yes, find A-1: 4 N 5 A = 0 2 0 3 2