1 2 (1) For A = [1 2 3 4 5 6 and B = 0 789 1 find the following if they exist. (each part 0.5 points) -2 (a) - 2A (b) A + 2B (c) AT (d) BT (e) AB (f) BA (2) A furniture maker has 12 units of wood and 21 hours of free time, in which he will make two models of chairs. He estimates that model I requires 4 units of wood and 7 hours of time, while model II requires 2 units of wood and 6 hours of time. The prices of models are $60 and $40, respectively. How many chairs of each model should the furniture maker assemble if he wishes to maximize his sale revenue? (a) Model the problem as an LP (define the decision variables, write the objective function and state the meaning of each constraints). (4 points) (b) Is (x = 0, x = 3) in the feasible region? (0.5 points) (c) Is (x= -1, x2 = 2) in the feasible region? (0.5 points) (d) Graphically solve the problem (find feasible region, draw 2 isoprofit lines and find the direction that objective function increases). What are the optimal solution and optimal value? (3 points) (e) Which constraints are binding at the optimal solution? (1 point) (f) Write the matrix form of the LP. (2 points) (3) Every LP with two variables must fall into one of the following cases: Case 1: The LP has an unique solution Case 2: The LP has multiple optimal solutions Case 3: The LP is Infeasible Case 4: The LP is unbounded Draw the feasible region and two isocost/profit lines and identify which of cases 1-4 apply to each of the following LPs. (each part 2 points) (a) max 4x1 + x (b) max -x + 3x2 (c) min- X2 s.t. 4x1+x2 8 s.t. s.t. 5x1 + 2x2 12 x 0 X 0 x + 2x 4 - x + x 4 X 0x 0 x + x 4 x - x > 5 X1 0 0