5 6x 4x 3 8y 3y 2 0 1

4 0 5

3 2 1 7 0 1 6 4

3

6 2 0580

3x 10y  15 5x 4y  22

8x 7y 4z  12 3x 5y 2z  20 5x 3y 3z  26

5 4 9 1 2 4 7 0 2 9 10 3  w,

13 1 4 16 21 10 7 8 4 3 5 3 2 12 1  5

0 1 2 1 2 2 1 3 2

4 3 2 8 1 10 16 2 12   0

1 0 0 2 1 0 3 2 1 9 2 0 x,

1 0 0 3 1 0 9 1 1 4 10 2  1

1 0 0 5 1 0 4 2 1 1 3 4

1 0 0 8 1 0 0 1 1 2 7 1   1 0

5x 4y  x y  2 22

2x 5y  2 3x 7y  1

0.3x 0.1y  0.13 0.2x 0.3y  0.25

0.2x 0.1y  0.4x 0.5y  0.07 0.01

x 2x 2y 4y   3 6

x 2x 2y 4y   3  6 x

x 2x x 2y y 3y z 2z 2z    7 4 3

x 2x x 2y y 3y z 2z 2z    4 24 20  x

2x y 2x 2y 2x y 2z  4  5 6z  2

x 2y 6z  2x 5y 15z  3x y 3z  1 4 6

2x 3y z  2x 3y 3z  4x 2y 3z  10 22 2

2x 3y 3z  3 6x 6y 12z  13 12x 9y z  2

2x 3x x y 2y 3y z 3z 2z z  w  2w  3w  6 9 11 14  2

x 4x 2x 2y 3y 4y 3z z z w  3  0 2w  0  3

x x 2x 2y y y z z 3z    3 3 10

x 3x x 3y y y z z 3z    2 6 2  x x

x y 2z  2x 3y z  5x 4y 2z  1 2 4

4x 4y 4z  5 4x 2y 8z  1 5x 3y 8z  6

2x y 9z  x 3y 4z  5x 2y z  8 15 17  4x

3x y 7z  5x 2y z  x y 4z  20 34 8

4x

1 y x 9   1 7 12 9 10

1 x 4 0 5 y    1 8 4 0 5 0

x 3 0 2 4 3 y 5 4y 2 6x    5x 1 0 2 4 3 16 44 2 6   1

9 0 6 4 3 1 2 7 1 5 4 0   x

2 3 2 5,

A   4 6 10 3 1 1 ,

A   5 7 11 4 2 2

A  6 5 7 ,

7 1 3 5  10 14 20 3 B

11 7 16 2 19 1  6 8 2 0 4 10  7

2  1 5 6 2 4 0  8  7 1 1 1 2 4

8 2 0 1 4 6 8 12 0  5  2 3 6 0 1 12 4 1 8  2

3  8 1 2 3 5 1 6  4 2 2 7 3 6  8

5  2 7 8 0 2 2 4  4 6 1 2 11 3  3

X  2A 3B 6X

6X  4A 3B

3X 2A  B

2A 5B  3X

A    2 3 2 5,

5 7 11 4 2 2 ,

A    5 7 11 4 2 2 , B   4 20 12 40

A  6 5 7 , B   1 4 8  A

1 5 6 2 4 0 6 4 2 0 8 0

1 2 5 4 6 06 4 2 0 8 0

1 2 5 4 6 0  6 2 8 4 0 0

1 0 0 3 2 0 2 4 3 4 0 0 3 3 0 2 1 2

4

2 6 1 0 4 3 2 1  2 0 4 4

3 1 4 1 2 0 1 3 21 5 0 3

4 11 12 1 7 3  3 2 5 2 6 2 3

2 4 3 2 10 2  1 5 3 1 2 2

MANUFACTURING A tire corporation has three factories, each of which manufactures two models of tires. The number of units of model produced at factory in one day is represented by in the matrix Find the production levels if production is decreased by 5%

MANUFACTURING A power tool company has four manufacturing plants, each of which produces three types of cordless power tools. The number of units of cordless power tool produced at plant in one day is represented by in the matrix Find the production levels if production is increased by 20%

MANUFACTURING An electronics manufacturing company produces three different models of headphones that are shipped to two warehouses. The number of units of model that are shipped to warehouse is represented by in the matrix

CELL PHONE CHARGES The pay-as-you-go charges (in dollars per minute) of two cellular telephone companies for calls inside the coverage area, regional roaming calls, and calls outside the coverage area are represented by Company A B Coverage area Each month, you plan to use 120 minutes on calls inside the coverage area, 80 minutes on regional roaming calls, and 20 minutes on calls outside the coverage area. (a) Write a matrix that represents the times spent on the phone for each type of call. (b) Compute and interpret the resul

A    4 7 1 2, 1

A    5 11 1 2, B

A   1 1 6 1 0 2 0 1 3

A   1 1 8 1 0 4 0 1 2 , B   2 3 2 1 1 2 1 2 1 2 1 2  A 

6 5 5 4 B

3 2 5 3

2 1 2 0 1 2 3 1 1

0 5 7 2 2 3 1 3 4   2 1

1 3 1 2 7 4 2 9 7

1 2 1 4 3 18 6 1 16   1

1 4 3 1 3 4 4 2 1 2 1

4 1 1 0 2 2 4 2 0 1 1 8 2 4 1

3 4 4 5 5 2 8 3

0.5 0.2 0.1 0.4

1.6 1.2 3.2 2.4

x 4y  2x 7y  8 5

5x y  9x 2y  13 24

3x 10y  5x 17y  8 13  5

4x 2y  19x 9y  10 47

0 1 2x 3x 1 3y 2y   2 0

5 6x 4x 3 8y 3y   2  0 1

25 0.3x 0.4x 0.7y 0.6y   10.2 7.6

3.5x 2.5x 4.5y 7.5y   8  25 0.

3x 2y z  x y 2z  5x y z  6 1 7

x 4y 2z  2x 9y 5z  x 5y 4z  12 25 10  3x

2x y 2z  x 4y z  y z  13 11 0  x

3x y 5z  x y 6z  8x 4y z  14 8 44  2x

x 2y  1 3x 4y  5

x 3y  6x 2y  23 18

98 6 5x 12 5 x 4 7y 12 7 y   6 5 17 5  x

5x 2x 10y y   7  98

3x 3y 4z  y z  4x 3y 4z  2 1 1  5x

x 3y 2z  2x 7y 3z  x y 3z  8 19 3  3x

8 2 5 4

9 7 11 4   8

50 10 30 5

14 12 24 15   50

2 7 1 4

3 5 6 4   2

3 2 1 2 5 8 1 0 6

8 6 4 3 5 1 4 9 2

2 2 1 0 1 1 0 0 3

0 0 1 1 1 1 2 2 3

4 2 1 1 3 1 1 2 0

1 2 5 2 3 1 1 0 3

2 6 5 4 0 3 1 2 4

1 4 0 1 1 1 4 2 1

1 1 2 2 2 2 4 0 1 4 3 0 0 1 1 0

1 4 2 0 2 1 3 2 1 4 3 4 2 1 0 2   1

3 0 6 0 0 8 1 3 4 1 8 4 0 2 2 1

5 0 3 1 6 1 4 6 0 1 5 0 0 2 1 3   3 0

5x 2y  11x 3y  6 23 10.

3x 8y  9x 5y  7 37

2x 3y 5z  4x y z  x 4y 6z  11

2x 3y 5z  4x y z  x 4y 6z  11

(1, 0) 2 4 2 468 6 8 y

(4, 0) 2 4 2 42 6 y

4 4 2 42 6 y

y 1 2 ( ( , 1 3 2( ( 1

1, 7, 3, 9, 3, 15 (4,

0, 5, 2, 6, 8, 1 1,

4, 0, 4, 4

2, 5, 6, 1

5 2, 3, 7 2, 1

0.8, 0.2, 0.7, 3.2 5

LOOK OUT BELOW 2 3 6 2 0 2 0 3 3 1

HEAD DUE WEST 1 3 1 2 7 4 2 9 7  3 1 2 7 4 2 9 7 

5 11 2 370 265 225 57 48 33 32 15 20 245 171 147

145 105 92 264 188 160 23 16 15 129 84 78 9 8 5 159 118 100 219 152 133 370 265 225 105 84 63

It is possible to find the determinant of a matrix.

a11 a21 a31 c1 a12 a22 a32 c2 a13 a23 a33 c3

Use the matrix capabilities of a graphing utility to find the inverse of the matrix What message appears on the screen? Why does the graphing utility display this message?

Under what conditions does a matrix have an inverse?

WRITING What is meant by the cofactor of an entry of a matrix? How are cofactors used to find the determinant of the matrix?

Three people were asked to solve a system of equations using an augmented matrix. Each person reduced the matrix to row-echelon form. The reduced matrices were and Can all three be right? Explain.

THINK ABOUT IT Describe the row-echelon form of an augmented matrix that corresponds to a system of linear equations that has a unique solution.

Solve the equation for 2  3 5 8   0