Copyright © 2011 Pearson, Inc. 7.3 Multivariate Linear Systems and Row Operations.

Copyright © 2011 Pearson, Inc.

7.3Multivariate

Linear Systems and Row

Operations

3 - 2 Copyright © 2011 Pearson, Inc.

What you’ll learn about

Triangular Forms for Linear Systems Gaussian Elimination Elementary Row Operations and Row Echelon Form Reduced Row Echelon Form Solving Systems with Inverse Matrices Applications

… 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.


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.


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:

2x +3y−z=−1−x+5y+3z=−103x−y−6z=5



Apply elementary row operations to find a row echelon

form of the augmented matrix.

2 3 −1 −1−1 5 3 −103 −1 −6 5

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−R21u ruuu

1 −5 −3 102 3 −1 −13 −1 −6 5

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−2R1 + R2u ruuuuuuuu

1 −5 −3 100 13 5 −213 −1 −6 5

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥−3R1 + R3u ruuuuuuuu

1 −5 −3 100 13 5 −210 14 3 −25

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

113

R2u ruuuu

2x +3y−z=−1−x+5y+3z=−103x−y−6z=5



1 −5 −3 10

0 1513

−2113

0 14 3 −25

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

−14R2 + R3u ruuuuuuuuuu

1 −5 −3 10

0 1513

−2113

0 0−3113

−3113

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

−1331

R3u ruuuuu

1 −5 −3 10

0 1513

−2113

0 0 1 1

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥



Convert the matrix to equations and solve by substitution.

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

The solution is 3,−2,1( ).

Solve the system:

2x +3y−z=−1−x+5y+3z=−103x−y−6z=5


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.


Invertible Square Linear System

Let A be the coefficient matrix of a system of n linear equations in n variables given by AX = B, where X is the n 1 matrix of variables and B is the n 1 matrix of numbers on the right-hand side of the equations. If A–1 exists, then the system of equations has the unique solution

X = A–1B.


Example Solving a System Using Inverse Matrices

Solve the system

2x −3y=02x−2y=10


Example Solving a System Using Inverse Matrices

Write the system as a matrix equation.

Let A =2 −32 −2

⎡

⎣⎢

⎤

⎦⎥, X =

xy

⎡

⎣⎢

⎤

⎦⎥, and B=

010

⎡

⎣⎢

⎤

⎦⎥.

Then A⋅X =2 −32 −2

⎡

⎣⎢

⎤

⎦⎥⋅

xy

⎡

⎣⎢

⎤

⎦⎥=

2x−3y2x−2y

⎡

⎣⎢

⎤

⎦⎥ so that

AX =B, where A is the coefficient matrix of the system.

A-1 exists since detA≠0. Use grapher to find

X =A-1B=1510

⎡

⎣⎢

⎤

⎦⎥. The solution of the system is (15,10).


Example Fitting a Parabola to Three Points

Determine a, b, and c so that −3, 32( ), 1, 4( ) and 5, 40( )

are on the graph of f x( ) =ax2 +bx+ c.





We must have f −3( ) =32, f 1( ) =4, and f 5( ) =40

f −3( ) =9a−3b+ c=32

f 1( ) =a+b+ c=4

f 5( ) =25a+5b+ c=40

A=9 −3 11 1 125 5 1

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥, X =

abc

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥, and B=

32440

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥





A grapher shows that

X =A−1B=2−35

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥.

Thus a=2, b=−3, and c=5.

The graph of the quadratic fucntion f x( ) =ax2 +bx+ c

contains the three points (–3, 32), (1, 4) and (5, 40).





Support Graphically

The figure shows a graph

of y1=2x2 −3x+5

superimposed on a scatterplot of the three points.The points appear to lieon the curve.


Quick Review

1. Find the amount of pure acid in 45L of a 58%

acid solution.

2. Find the amount of water in 30 L of a 28%

acid solution.

3. Is the point (0, −1) on the graph of the function

f (x) =x3 −4x−1?4. Solve for x in terms of the other variables:x + z + w = 2

5. Find the inverse of the matrix 2 10 3

⎡

⎣⎢

⎤

⎦⎥.


Quick Review Solutions

1. Find the amount of pure acid in 45L of a 58%

acid solution. 26.1 L

2. Find the amount of water in 30 L of a 28%

acid solution. 21.6 L

3. Is the point (0, −1) on the graph of the function

f (x) =x3 −4x−1? yes

4. Solve for x in terms of the other variables:x + z + w = 2 x=2−z−w

5. Find the inverse of the matrix 2 10 3

⎡

⎣⎢

⎤

⎦⎥

1/2 −1/ 60 1/ 3

⎡

⎣⎢

⎤

⎦⎥.

Copyright © 2011 Pearson, Inc. 7.3 Multivariate Linear Systems and Row Operations.

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Transcript of Copyright © 2011 Pearson, Inc. 7.3 Multivariate Linear Systems and Row Operations.