example 1 Matrix Solution of Linear Systems

Chapter 7.2

Use matrix row operations to solve the system of equations

2 3 1

2 3

3 9

x y z

x y z

x y z

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Use matrix row operations to solve the system of equations

2 3 1

2 3

3 9

x y z

x y z

x y z

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Use matrix row operations to solve the system of equations

2 3 1

2 3

3 9

x y z

x y z

x y z

2 3

2 3 1

3 9

x y z

x y z

x y z

R1 R2

2 3 1 1

1 1 2 3

3 1 1 9

1 1 2 3

2 3 1 1

3 1 1 9

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Use matrix row operations to solve the system of equations

2 3 1

2 3

3 9

x y z

x y z

x y z

2 3

2 3 1

3 9

x y z

x y z

x y z

R1 R2

2 3 1 1

1 1 2 3

3 1 1 9

1 1 2 3

2 3 1 1

3 1 1 9

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Use matrix row operations to solve the system of equations

2 3 1

2 3

3 9

x y z

x y z

x y z

2 3

2 3 1

3 9

x y z

x y z

x y z

R1 R2

2 3 1 1

1 1 2 3

3 1 1 9

1 1 2 3

2 3 1 1

3 1 1 9

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2 3

3 5

4 7 18

x y z

y z

y z

-1 R2 R2 1 1 2 3

0 1 3 5

0 4 7 18

-2 R1 + R2 R2-3 R1 + R3 R3

2 3

3 5

4 7 18

x y z

y z

y z

1 1 2 3

0 1 3 5

0 4 7 18

1 1 2 3

2 3 1 1

3 1 1 9

2 2 4 6

2 3 1 1

0 1 3 5

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2 3

3 5

4 7 18

x y z

y z

y z

-1 R2 R2 1 1 2 3

0 1 3 5

0 4 7 18

-2 R1 + R2 R2-3 R1 + R3 R3

2 3

3 5

4 7 18

x y z

y z

y z

1 1 2 3

0 1 3 5

0 4 7 18

1 1 2 3

2 3 1 1

3 1 1 9

2 2 4 6

2 3 1 1

0 1 3 5

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2 3

3 5

4 7 18

x y z

y z

y z

-1 R2 R2 1 1 2 3

0 1 3 5

0 4 7 18

-2 R1 + R2 R2-3 R1 + R3 R3

2 3

3 5

4 7 18

x y z

y z

y z

1 1 2 3

0 1 3 5

0 4 7 18

1 1 2 3

2 3 1 1

3 1 1 9

2 2 4 6

2 3 1 1

0 1 3 5

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2 3

3 5

4 7 18

x y z

y z

y z

-1 R2 R2 1 1 2 3

0 1 3 5

0 4 7 18

-2 R1 + R2 R2-3 R1 + R3 R3

2 3

3 5

4 7 18

x y z

y z

y z

1 1 2 3

0 1 3 5

0 4 7 18

1 1 2 3

2 3 1 1

3 1 1 9

3 3 6 9

3 1 1 9

0 4 7 18

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2 3

3 5

4 7 18

x y z

y z

y z

-1 R2 R2 1 1 2 3

0 1 3 5

0 4 7 18

-2 R1 + R2 R2-3 R1 + R3 R3

2 3

3 5

4 7 18

x y z

y z

y z

1 1 2 3

0 1 3 5

0 4 7 18

1 1 2 3

2 3 1 1

3 1 1 9

3 3 6 9

3 1 1 9

0 4 7 18

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2 3

3 5

4 7 18

x y z

y z

y z

-1 R2 R2 1 1 2 3

0 1 3 5

0 4 7 18

-2 R1 + R2 R2-3 R1 + R3 R3

2 3

3 5

4 7 18

x y z

y z

y z

1 1 2 3

0 1 3 5

0 4 7 18

1 1 2 3

2 3 1 1

3 1 1 9

3 3 6 9

3 1 1 9

0 4 7 18

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2 3

3 5

4 7 18

x y z

y z

y z

-1 R2 R2 1 1 2 3

0 1 3 5

0 4 7 18

-2 R1 + R2 R2-3 R1 + R3 R3

2 3

3 5

4 7 18

x y z

y z

y z

1 1 2 3

0 1 3 5

0 4 7 18

1 1 2 3

2 3 1 1

3 1 1 9

3 3 6 9

3 1 1 9

0 4 7 18

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2 3

3 5

4 7 18

x y z

y z

y z

-1 R2 R2 1 1 2 3

0 1 3 5

0 4 7 18

-2 R1 + R2 R2-3 R1 + R3 R3

2 3

3 5

4 7 18

x y z

y z

y z

1 1 2 3

0 1 3 5

0 4 7 18

1 1 2 3

2 3 1 1

3 1 1 9

3 3 6 9

3 1 1 9

0 4 7 18

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2 3

3 5

4 7 18

x y z

y z

y z

-1 R2 R2 1 1 2 3

0 1 3 5

0 4 7 18

-2 R1 + R2 R2-3 R1 + R3 R3

2 3

3 5

4 7 18

x y z

y z

y z

1 1 2 3

0 1 3 5

0 4 7 18

1 1 2 3

2 3 1 1

3 1 1 9

3 3 6 9

3 1 1 9

0 4 7 18

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2 3

3 5

4 7 18

x y z

y z

y z

-1 R2 R2 1 1 2 3

0 1 3 5

0 4 7 18

-2 R1 + R2 R2-3 R1 + R3 R3

2 3

3 5

4 7 18

x y z

y z

y z

1 1 2 3

0 1 3 5

0 4 7 18

1 1 2 3

2 3 1 1

3 1 1 9

3 3 6 9

3 1 1 9

0 4 7 18

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38

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2 3

3 5

19 38

x y z

y z

z

-4 R2 + R3 R3

2 3

3 5

2

x y z

y z

z

R3 R3119

1 1 2 3

0 1 3 5

0 0 19 38

1 1 2 3

0 1 3 5

0 0 1 2

1 1 2 3

0 1 3 5

0 4 7 18

0 4 12 20

0 4 7 18

0 0 19 38