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CHAPTER 8

MATRICES AND

DETERMINANTS

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8.1 Matrix Solutions to LinearSystems

Objectives

Write the augmented matrix for a linearsystem

Perform matrix row operations

Use matrices & Gaussian elimination to solvesystems

Use matrices & Gauss-Jordan elimination tosolve systems

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Solving a system of 3 equationswith 3 variables

Graphically, you are attempting to findwhere 3 planes intersect.

If you find 3 numeric values for (x,y,z), thisindicates the 3 planes intersect at thatpoint.

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Representing a system ofequations in a matrix

If a linear system of 3 equations involved 3variables, each column represents the differentvariables & constant, and each row represents a

separate equation. Example: Write the following system as a matrix

2x + 3y 3z = 7

5x + y 4z = 2

4x + 2y - z = 6

6124

2415

7332

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Solving linear systems usingGaussian elimination

Use techniques learned previously tosolve equations (addition & substitution) tosolve the system

Variables are eliminated, but each columnrepresents a different variable

Perform addition &/or multiplication to

simplify rows. Have one row contain 2zeros, a one, and a constant. This allowsyou to solve for one variable.

Work up the matrix and solve for the

remaining variables.

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A matrix should look like this afterGaussian elimination is applied

3

22

111

100

10

1

k

kz

kzy

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Examples 3 & 4 (p. 541-543)

These examples outline a stepwiseapproach to use Gaussian elimination.

One row (often the bottom row) willcontain (0 0 0 1 constant) using thismethod.

After the last variable is solved in thebottom equation, substitute in for thatvariable in the remaining equations.

GO THROUGH EXAMPLES CAREFULLY

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ues on: us e as rowcontain (0 0 0 1 constant)?

Yes, in Gaussian elimination, but there are other

options. Look again at example 3, page 539.

If you multiply the first row by (-1) and add it tothe 2nd row, the result is (-2 0 0 -12)

What does this mean? -2x = -12, x=6 By making an informed decision as to what

variables to eliminate, we solved for a variablemuch more quickly!

Next, multiply row 1 by (-1) & add to 3rd row: (-22 0 6). Recall x=-6, therefore this becomes-2(6)+2y=6, y=3

Now knowing x & y, solve for z in any row (row 2?)

6 + 3 + 2z = 19, z=5 Solution: (6,3,5)

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If this method is quicker, why wouldwe use Gaussian elimination?

Using a matrix to solve a system byGaussian elimination provides a standard,programmable approach.

When computer programs (may becontained in calculators) solve systems.This is the method utilized!

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8.2 Inconsistent & DependentSystems & Their Applications

Objectives

Apply Gaussian elimination to systemswithout unique solutions

Apply Gaussian elimination to systems withmore variables than equations

Solve problems involving systems without

unique solutions.

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How would you know, with

Gaussian elimination, that there are

no solutions to your system?

When reducing your matrix (attempting tohave rows contain only 0s, 1s & the

constant) a row becomes 0 0 0 0 k

What does that mean? Can you have 0times anything equal to a non-zeroconstant? NO! No solution!

Inconsistent system no solution

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Graphically, what is happening withan inconsistent system?

Recall, with 3 variables, the equationrepresents a plane, therefore we areconsidering the intersection of 3 planes.

If a system is inconsistent, 2 or more ofthe planes may be parallel OR 2 planescould intersect forming 1 line and a

different pair of planes intersect at adifferent line, therefore there is nothing incommon to all three planes.

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Could there be more than onesolution?

Yes! If the planes intersect to form a line,rather than a point, there would beinfinitely many solutions. All pts. lying on

the line would be solutions.

You cant state infinitely many points,

so you state the general form of all

points on the line, in terms of one ofthe variables.

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What if your system has 3 variablesbut only 2 equations?

Graphically, this is the intersection of 2planes.

2 planes cannot intersect in 1 point, rather

they intersect in 1 line. (or are parallel,thus no solution)

The solution is all points on that line.

The ordered triple is represented as one ofthe variables (usually z) and the other 2 asfunctions of that variable: ex: (z+2,3z,z)

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Dependent system

Notice when there were infinitely manysolutions, two variables were stated interms of the 3rd. In other words, the x & y

values are dependent on the valueselected for z.

If there are infinitely many solutions, the

system is considered to be dependent.

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8.3 Matrix Operations & TheirApplications

Objectives

Use matrix notation

Understand what is meant by equal matrices

Perform scalar multiplication

Solve matrix equations

Multiply matrices

Describe applied situations with matrixoperations

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What is a matrix?

A set of numbers in rows & columns

m x n describes the dimensions of the matrix (mrows & n columns)

The matrix is contained within brackets.

Example of a 3 x 3 matrix:

830

341524

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Add & Subtract matrices

Only if they are of the same dimensions

Add (or subtract) position by position (i.e.the term in the 3rd row 2nd column of the 1stmatrix + the term in the 3rd row 2nd columnof the 2nd matrix)

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Scalar Multiplication

When the multiplier is on the outside of thematrix, every term in the matrix is multiplied bythat constant.

You can combine multiplication and addition ofmatrices (PRS on next slide)

820

164

25

414

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Find 3A + B if

14

32,

32

11BA

810

05)4

96

33

)3

121869)2

46

23)1

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Multiplication of matrices

When multiplying matrices, order matters!

The first row of the 1st matrix is multiplied(term by term) by the 1st column of the 2ndmatrix. The sum of these products is thenew term for the element in the 1st row, 1stcolumn of the new product matrix.

The # columns in 1st matrix MUST equal #rows in 2ndmatrix (otherwise terms wontmatch up in the term by term

multiplication)

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Multiply:

1)1st row of 1st by 1st column of 2nd (3(3)+4(4)=25)

2) 1st row of 1st by 2nd column of 2nd (3(1)+4(-2)=-5)

3)1st row of 1st by 3rd column of 2nd (3(-1)+4(1)=1)

4)1st row of 1st by 4th column of 2nd (3(2)+4(0)=6)

5)2nd row of 1st by 1st column of 2nd (-2(3)+5(4)=14)

6)2nd row of 1st by 2nd column of 2nd (-2(1)+5(-2)=-12)

7)2nd row of 1st by 3rd column of 2nd (-2(-1)+5(1)=7)

8)2nd row of 1st by 4th column of 2nd (-2(2)+5(0)=-4)

(continue on next slide)

0124

2113

52

43

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New matrix is created

Multiplying 1st row by 1st column fills position 11,multiplying 1st row by 2nd column fills position 12,etc

Result:

Compare this matrix with the results from theprevious slide

47121461525

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8.4 Multiplicative Inverses ofMatrices & Matrix Equations

Objectives

Find the multiplicative inverse of a squarematrix

Use inverses to solve matrix equations

Encode & decode messages

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What is the multiplicative inverse ofa matrix?

Its the matrix that you must multiply another

matrix by to result in the identity matrix.

What is the identity matrix? Its the matrix that

you would multiply another matrix by that wouldnot change the value of the original matrix.

The identity matrix of a 3x3 matrix is:

100

010001

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How do we find the multiplicativeinverse of a matrix?

Example 2, p. 574, outlines the steps. In general,if its a 2x2 matrix, youre finding the value of 4

unknowns using 2systems of equations, each with

2 variables. Example on next slide for finding a multiplicative

inverse of a matrix ( ), which means themultiplicative inverse, NOT 1/A.

1A

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Find the multiplicativeinverse of

Multiplicative inverse is:

22

15

122

05022

15

10

01

22

15

db

dbca

ca

dc

ba

40/9,8/1,18

4/1,4/1,28

dbb

caa

40/94/18/14/1

Quick method for finding the

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Quick method for finding themultiplicative inverse of a 2x2

matrix

If

ac

bd

bcadA

dc

baA

1,

1

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What if the matrix is greater than a2x2?

Form an augmented matrix where A isthe original matrix & I is the identity matrix.

Perform elimination and substitution until the

original matrix (A) appears as the identity matrix. The resulting matrix left where the identity matrix

began is the multiplicative inverse matrix.

IA

)(1

A

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8.5 Determinants & Cramers Rule

Objectives

Evaluate a 2nd-order determinant

Solve a system of linear equations in 2

variables using Cramers rule

Evaluate a 3rd-order determinant

Solve a system of linear equations in 3

variables using Cramers rule Use determinants to identify inconsistent &

dependent systems

Evaluate higher-order determinants

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Determinant of a 2x2 matrix

If A is a matrix, the determinant is A

234)2()5(3

54

23,

54

23

AA

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When are determinants useful?

They can be used to solve a system of equations

Cramers Rule

22

11

22

11

22

11

22

11

222

111

,

ba

ba

ca

ca

y

ba

ba

bc

bc

x

cybxa

cybxa

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Finding a determinant of a 3x3matrix

More complicated, but it can be done!

Its often easier to pick your home row/column(the one with the multipliers) to be a row/columnthat has one or more zeros in it.

22

11

333

11

233

22

1

333

222

111

cb

cb

acb

cb

acb

cb

acbacba

cba

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Determinants can be used to solve a

linear system in 3 variablesCRAMERS RULE

DDz

DDy

DDx zyx ,,

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What is D and

D is the determinant that results from thecoefficients of all variables.

is the determinant that results when each xcoefficient is replaced with the given constants.

is the determinant that results when each y

coefficient is replaced with the given constants.

is the determinant that results when the z

coefficients are replaced with the given constants.

zyx DDD ,,

xD

yD

zD

Find z given

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Find z, given2x + y = 7

-x + 3y + z = 5

3x + 2y 4z = 10

1023

531

712

423

131

012

)2(

423

131

012

1057

131

012

)1

423

131

712

1023

531

712

)4(

423

131012

1023

531

712

)3