Quiz 2 Answer key - University of Pittsburgh
[Pages:2]Fall 2013
Quiz 2
Answer key
Math 0400
1. [5 points] Using the Gauss-Jordan elimination method solve the system of linear equations
x-y+z = 3 x + y - z = -1
y+z = 4
Solution: The augmented matrix is
1 -1 1 3
1 -1 1 3
1 0 0 1
1 1 -1 -1 -R-2---R1 0 1 -1 -2 -R-1-+-R2 0 1 -1 -2
01 1 4
R2/2
01 1 4
R3-R2
00 2 6
1 0 0 1
--R-3/-2 0 1 0 1
R2+R3
0013
Answer: x = 1, y = 1, z = 3
2. [5 points] Using the Gauss-Jordan elimination method determine whether the system has a solution and find the solution or solutions, if they exist.
3x + 3z = 6 2y = -2
2x + 3y + 2z = 1
Solution: The augmented matrix is
3 0 3 6
1 0 1 2
1 0 1 2
1 0 1 2
0 2 0 -2 -R-2-/2 0 1 0 -1 -R-3--2-R1 0 1 0 -1 -R-3--3-R2 0 1 0 -1
2 3 2 1 R1/3 2 3 2 1
0 3 0 -3
000 0
The given system is equivalent to the system x+z = 2 y = -1
Letting z = t, where t is a parameter, we see that the infinitely many solutions are given by x = 2 - t, y = -1, z = t
Another form of the solution: Letting x = t, we get infinitely many solutions x = t, y = -1, z =2-t
Answer: Either x = 2 - t, y = -1, z = t or x = t, y = -1, z = 2 - t.
1
3. The Coffee Shoppe sells a coffee blend made from two coffees, one costing $5/lb and the other costing $6/lb. If the blended coffee sells for $5.60/lb, find how much of each coffee is used to obtain the desired blend. Assume that the weight of the blended coffee is 100 lb.
(a) [1 points] Formulate the problem as a system of linear equations (b) [2 points] Express the system as a matrix equation (c) [2 points] Solve the system using the inverse matrix
Solution: (a) Denote by x the weight in lb of the first cofee in the blend and by y the weight of the second cofee. Then we obtain a system of two equations with two unknowns:
x + y = 100
or
5x + 6y = 5.6 ? 100
x + y = 100 5x + 6y = 560
(b) AX = B, where A =
1 5
1 6
, X=
x y
, B=
100 560
(c) A-1: ad - bc = 6 - 5 = 1. Then A-1 =
6 -1 -5 1
and
X=
x y
= A-1B =
6 -1 -5 1
100 560
=
600 - 560 -500 + 560
=
40 60
Answer: The weight of coffee that costs $5/lb is 40 lbs and the weight of coffee that costs $6/lb is 60 lbs.
bonus problem [5 points extra] Find a matrix A such that AB = C
if B =
2 4
1 3
,C=
-2 -3 62
Solution:
ABB-1 = CB-1, A = CB-1, B-1 = 1 2
3 -1 -4 2
=
3/2 -2
-1/2 1
,
A = CB-1 =
-2 -3 62
3/2 -1/2 -2 1
=
-3 + 6 1 - 3 9 - 4 -3 + 2
=
3 -2 5 -1
Note: A = B-1C =
3/2 -1/2 -2 1
-2 -3 62
=
-3 - 3 -9/2 - 1 4+6 6+2
=
-6 -11/2 10 8
Answer: A =
3 5
-2 -1
.
2
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