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Unformatted text preview: Math 201 — Test 3 Section: Name: Show appropriate Work to receive credit on all problems and place your answer in the blank provided. Point value of each problem is
listed in the problem statement. Calculators are not allowed on this exam. 1 3 5 2
1. If matrix A = V 2 1 4 3 find an and the size of A: (2 points each) an: 7 2 0 8
SizeofA: 3 2
(4 points each) 1 4 —1 2 —2 —1 O 1 0
2. GivenA= [ :l , B = [ C = [ 1 0 I] , and I=[O 1], find each of the following: a.) AT b.) A + 2B c.) BC d.) B1 3. Write a 2 X 2 matrix with entries au = i + j. (4 points) 4. Complete each of the following: (a: 3 points, b: 5 points)
a.) If A is a 3 X 4 matrix and B is a 4 X 6 matrix, then the size of AB is b.) If A is a 6 X 7 matrix and B is a 6 X 3 matrix, is the matrix multiplication AB possible? Explain: 5. Solve each system of equations below using an augmented matrix and matrix reduction. State your answer in terms tutwwmlm‘mwmsmvmwmnaaaaaaaawnumouam'knckwwmw of a parameter if necessary. (6 points each) a.) 2x + y = 8
x — 2y = —6 b.) X + y — 2Z = 0 (You may work only with the coefficient matrix ifyou wish.)
2x + By — 52 = O 6. Classify each of the following as True or False by circling the appropriate letter. (2 points each) 1 0 3 0
T F a. ) If a reduced matrix for a linear system is 0 1 2 0 , then the system has no solution.
0 0 0 0
T F b.) A homogeneous system of three linear equations with four unknowns has exactly one solution, the trivial solution. 7 a.) Write the system of equations below as a matrix equation in the form AX = B: (2 points) x+3y=2 x + 2y = 5 Matrix Equation: h.) Find the inverse, that is A“, for the coefficient matrix A from part 7a above. (Spoints) c.) Use A“ to solve the system given in 7a. (4 points) 1 Solution: 8. The region indicated in the diagram below is best described by: (Circle your selection.) (5 points) 2x 2x <2x 2x 2x
A) y B) y C) y D) y E) x+y>1 x+y 1 x+y>l x+y>1 x+y 1 9. Maximize Z = 4x + y subject to constraints —x + y 5 1, 2x + y 5 10, x30, y30using each method below. a.) Linear programming. Graph the constraints, the feasible region, and complete each portion of the work
identified below: (13 points) Indicate whether you left the feasibleregion shaded or open. (Circle one.) y Coordinates of vertices (corners) of feasible region: Process to maximize Z : N Value of Z and coordinates at the maximum: (Problem restatement: Maximize Z = 4x +y subject to constraints —x + y ’5 1, 2x + y _<_ 10 , x30, y_>_0) b.) The Simplex Method: (13 points) Write constraints stated as equations with slack variables here: (You may work with x and y or with x. and x; in place
ofx & y if you wish) Initial Simplex Table: and continue: Result: 10. Problem Statement: A manufacturer produces two products, product A and product B. Both products require processing on Machine I and
II. The number of hours needed to produce one unit is given in the chart below. Machine I is available for at most 1140
hours per month and Machine 11 is available for at most 1200 hours per month. The profit made on product A is $15/unit and the profit made on product B is $30/unit. Find the production level that will maximize profit and find the
maximum profit. Identify variables and set up the objective function and the constraints only. Do not solve. (9 points) Machine I Machine II Profit
Product A 2 hrs 3 hrs
Product B 6 hrs 4 hrs
Unknowns (be precise): Constraints:
x :
y = — Objective Function: :u'nmmuwxmrmmwﬂuauelwwmaluminum;minAvwmnadnaaaanMmauam‘kncwuuWugw yarn»:mumxmrmmwﬂnaew‘wwmmrmwm;IotaAvmammalianaaanmanumauamkmckwwmw umvmnxxm.wmm{MannManwnumauemknckwwm‘ ...
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 Spring '08
 SMITH

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