Quiz 2 Answer key - University of Pittsburgh

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

.

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