Gauss-Jordan, Matrix Inverses, Production Matrices & Echelon Method

(See attached file for full problem description)

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1. What is true about the number of solutions to a system of m linear equations in n unknowns if m = n? If m < n? If m > n?

Solve each system by the echelon method.

3. 2x + 3y = 10
-3x + y = 18

5. 2x - 3y + z = -5
x + 4y + 2z = 13
5x + 5y + 3z = 14

Solve each system by the Gauss-Jordan method.

7. 2x + 4y = -6
-3x + 5y = 12

9. x - y + 3z = 13
4x + y + 2z = 17
3x + 2y + 2z = 1

11. 3x - 6y + 9z = 12
-x + 2y - 3z = -4
x + y + 2z = 7

Find the size of each matrix, find the values of any variables, and identify any square, row, or column matrices.

13.

[ 2 x ] [ a -1 ]
[ y 6 ] = [ 4 6 ]
[ 5 z ] [ p 7 ]

15.

[ a + 5 3 + b 6 ] [-7 b+2 2k-3]
[ 4c 2 + d -3 ] = [ 3 2d -1 4l ]
[ -1 4p q - 1 ] [ m 12 8 ]

Given the matrices:

A = [4 10]
[-2 -3]
[6 9]

B = [2 3 -2]
[2 4 0]
[0 1 2]

C = [5 0]
[-1 3]
[4 7]

D = [6]
[1]
[0]

E = [1 3 -4]

F = [-1 4]
[3 7]

G = [2 5]
[1 6]

Find each of the following, if it exists.

17. 2G - 4F
19. B - A
21. AF
23. DE
25. BD
-1
27. F
-1
29. (A + C)

Find the inverse of each matrix that has an inverse.

31. [-4 2]
[0 3]

33. [6 4]
[3 2]

35. [2 0 4]
[1 -1 0]
[0 1 -2]

37. [ 2 3 5]
[-2 -3 -5]
[1 4 2]

Solve the matrix equation AX = B for X using the given matrices.

39.
[1 2]
A = [2 4]

B = [ 5 ]
[ 10 ]

41.

A =

[2 4 0]
[1 -2 0]
[0 0 3]

B =

[72]
[-24]
[48]

Solve each system of equations by inverses.

43. 5x + 10y = 80
3x = 2y = 120

45. x + 4y - z = 6
2x - y + z = 3
3x + 2y + 3z = 16

Find each production matrix, given the following input-output and demand matrices.

47.

A = [.2 .1 .3]
[.1 0 .2]
[0 0 .4]

D = [500]
[200]
[100]

In Exercises 49/51 below, write a system of equations and solve.

49. An office supply manufacturer makes two kinds of paper clips, standard and extra large. To make 1000 standard paper clips requires ¼ hour on a cutting machine and ½ hour on a machine that shapes the clips. One thousand extra large paper clips require 1/3 hour on each machine. The manager of paper clip production has 4 hours per day available on the cutting machine and 6 hours per day on the shaping machine. How many of each kind of clip can he make?

51. The Waputi Indians make woven blankets, rugs, and skirts. Each blanket requires 24 hours for spinning the yarn, 4 hours for dyeing the yarn, and 15 hours for weaving. Rugs require 30, 5, and 18 hours and skirts 12, 3, and 9 hours, respectively. If there are 306, 59, and 201 hours available for spinning, dyeing, and weaving, respectively, how many of each item can be made? (Hint: Simplify the equations you write, if possible, before solving the system.)

53. The New York Stock Exchange reports in daily newspapers give the dividend, price-to-earnings ratio, sales (in hundreds of sales), last price, and change in price for each company. Write the following stock reports as a 4 X 5 matrix: American Telephone & Telegraph: .14, 11, 333,675, 20.13, +1.88, General Electric: .64, 39, 390,591, 47.81, +4.06; Lucent: .08, 41, 436,351, 15.19, +1.88; Sara Lee: .54, 17, 27,077, 23.13, -1.50.

55. An economy depends on two commodities, goats and cheese. It takes 2/3 of a unit of goats to produce 1 unit of cheese and ½ unit of cheese to produce 1 unit of goats.

a. Write the input-output matrix for this economy
b. Find the production required to satisfy a demand of 400 units of cheese and 800 units of goats.

57. The activities of a grazing animal can be classified roughly into three categories: grazing, moving, and resting. Suppose horses spend 8 hours grazing, 8 moving, and 8 resting; cattle spend 10 grazing, 5 moving and 9 resting; sheep spend 7 grazing, 10 moving, and 7 resting; and goats spend 8 grazing, 9 moving, and 7 resting. Write this information as a 4 X 3 matrix.

65. Suppose 20% of the boys and 30% of the girls in a high school like tennis, and 60% of the boys and 90% of the girls like math. If 500 students like tennis and 1500 like math, how many boys and girls are in the school? Find all possible solutions.
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(See attached file for full problem description)