Multivariate Linear Systems and Row Operations.

Lesson7.3

Multivariate Linear Systems and

Row Operations

Precalculus

Quick Review

31. Find the exact value (w/o a calculator) of cos .

42. Find exact value (w/o a calculator) of sin .

3. Write the expression in factored form involving

one trigonometric function only: c

2os sin 1

2 35. Find the inverse of the matrix .

6 4

x x

2

2

sin(0) 0 sin 0

2cos (1 cos ) 1x x

4 3

6 2

1

10

325 10

3 15 5

sin(2 ) 0

2cos cos 2x x (cos 2)(cos 1) x x

What you’ll learn about

Triangular Forms for Linear SystemsGaussian EliminationElementary Row Operations and Row Echelon FormReduced Row Echelon FormSolving Systems with Inverse MatricesApplications

… and whyMany applications in business and science are

modeled by systems of linear equations in three or more variables.

Equivalent Systems of Linear Equations

The following operations produce an equivalent system of linear equations.1. Interchange any two equations of the

system.2. Multiply (or divide) one of the equations by

any nonzero real number.3. Add a multiple of one equation to any other

equation in the system.

Transforming a system to triangular form is called Gaussian elimination.

Using Gaussian Elimination

Solve the system of equations using Gaussian elimination. 2 3 7

2 3 14

4 2 3

x y z

x y z

x y z

Mult. 1st equation by -2 and add to 2nd equation, replacing the 2nd.

2 3 7

7 7 28

4 2 3

x y z

y z

x y z

Mult. 1st equation by -4 and add to 3rd equation, replacing the 3rd.

2 3 7

7 7 28

7 10 31

x y z

y z

y z

Mult. 2nd equation by -1 and add to 3rd equation, replacing the 3rd.

2 3 7

7 7 28

1

x y z

y z

z

Triangular form makes the solution easy to read.

Solution: 2, 3, 1x y z

Using Gaussian Elimination

Solve the system of equations using Gaussian elimination.

2 3 4

4 2 3 2

6 7 7 5

x y z

x y z

x y z

Mult. 1st equation by -2 and add to 2nd equation, replacing the 2nd.

2 3 4

8 5 6

6 7 7 5

x y z

y z

x y z

Mult. 1st equation by 3 and add to 3rd equation, replacing the 3rd.

2 3 4

8 5 6

16 10 17

x y z

y z

y z

Mult. 2nd equation by 2 and add to 3rd equation, replacing the 3rd.

2 3 4

8 5 6

0 5

x y z

y z

No Solution

Row Echelon Form of a Matrix

A matrix is in row echelon form if the following conditions are satisfied.1. Rows consisting entirely of 0’s (if there are

any) occur at the bottom of the matrix.2. The first entry in any row with nonzero

entries is 1.3. The column subscript of the leading 1 entries

increases as the row subscript increases.

Another way to phrase parts 2 and 3 is to say that the leading 1’s move to the right as we move down the rows.

Elementary Row Operations on a Matrix

A combination of the following operations will transform a matrix to row echelon form.1. Interchange any two rows.2. Multiply all elements of a row by a nonzero

real number.3. Add a multiple of one row to any other row.

Example Finding a Row Echelon Form

Solve the system:

2 3 1

5 3 10

3 6 5

x y z

x y z

x y z

The augmented matrix of this system of equations is:

2 3 1 1

1 5 3 10

3 1 6 5


21RIndicates interchanging the ith and jth row of the matrix.

12RIndicates multiplying the ith row by −2.

21

Apply elementary row operations to find

a row echelon form of the augmented matrix.

2 3 1 1 1 5 3 10

1 5 3 10 2 3 1 1

3 1 6 5 3 1 6 5

R

��

1 2

1 5 3 10

2 0 13 5 21

3 1 6 5

R R

��


1 33R R Indicates multiplying the ith row by −3 and adding it to the jth row.

1 2 1 3

Apply elementary row operations to find

a row echelon form of the augmented matrix.

1 5 3 10 1 5 3 10

2 0 13 5 21 3 0 13 5 21

3 1 6 5 0 14 3 25

R R R R

��


2 2 3

1 5 3 10 1 5 3 10

1 5 21 5 210 1 14 0 1

13 13 13 13 130 14 3 25 31 31

0 013 13

R R R

��

3

1 5 3 10

13 5 210 1

31 13 130 0 1 1

R

��

Apply elementary row operations to find a row echelon form

of the augmented matrix.


Convert the matrix to equations and solve by substitution.

1; 5 /13 21/13 so 2; 10 3 10 so 3.z y y x x

3

1 5 3 10 1 5 3 10

5 21 13 5 210 1 0 1

13 13 31 13 1331 31 0 0 1 1

0 013 13

R

��

Apply elementary row operations to find a row echelon form

of the augmented matrix.

The solution is 3, 2,1 .

Reduced Row Echelon Form

If we continue to apply elementary row operations to a row echelon form of a matrix, we can obtain a matrix in which every column that has a leading 1 has 0’s elsewhere. This is the reduced echelon form.

Example Reduced Row Echelon Form

1 2 3 3

4 1 2 2

2 3 8 5

Apply elementary row operations to find a reduced row echelon

form of the augmented matrix.

1 2

1 2 3 3

4 0 7 14 14

2 3 8 5

R R

��

1 3

1 2 3 3

2 0 7 14 14

0 7 14 1

R R

�� 3 2

1 2 3 3

0 7 14 14

0 0 0 13

R R

��

Example Reduced Row Echelon FormApply elementary row operations to find a reduced row echelon


1 2 3 3

0 7 14 14

0 0 0 13

3

1 2 3 31

0 7 14 14 13

0 0 0 1

R

��

2

1 2 3 31

0 1 2 2 7

0 0 0 1

R

��3 2

1 2 3 3

2 0 1 2 0

0 0 0 1

R R

��

Example Reduced Row Echelon FormApply elementary row operations to find a reduced row echelon


1 2 3 3

0 1 2 0

0 0 0 1

2 1

1 0 1 3

2 0 1 2 0

0 0 0 1

R R

�� 3 1

1 0 1 0

3 0 1 2 0

0 0 0 1

R R

��

1 2 3 3

Reduced Row Echelon form of 4 1 2 2

2 3 8 5

1 0 1 0

yields 0 1 2 0

0 0 0 1

Homework:

Text pg602 Exercises #4, 6, 10, 12, 18

Multivariate Linear Systems and Row Operations.

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