p-bltzmc08 805-872-hr 21-11-2008 13:26 page 827 matrix...

Section 8.3 Matrix Operations and Their Applications 827

Objectives

� Use matrix notation.

� Understand what is meant byequal matrices.

� Add and subtract matrices.

� Perform scalar multiplication.

� Solve matrix equations.

� Multiply matrices.

� Model applied situations withmatrix operations.

Matrix Operations and Their Applications

Turn on your computer and read youre-mail or write a paper. When you

need to do research, use the Internetto browse through art museums and

photography exhibits. When youneed a break, load a flight simula-tor program and fly through aphotorealistic computer world. Asdifferent as these experiences maybe, they all share one thing—you’re looking at images based onmatrices. Matrices have applica-tions in numerous fields, including

the new technology of digital photo-graphy in which pictures are represented

by numbers rather than film. In thissection, we turn our attention to matrix

algebra and some of its applications.

Notations for MatricesWe have seen that an array of numbers, arranged in rows and columns and placed inbrackets, is called a matrix. We can represent a matrix in two different ways.

• A capital letter, such as or can denote a matrix.

• A lowercase letter enclosed in brackets, such as that shown below, can denotea matrix.

A general element in matrix is denoted by This refers to the element inthe row and column. For example, is the element of located in thethird row, second column.A matrix of order has rows and columns. If a matrix has

the same number of rows as columns and is called a square matrix.

Matrix Notation

Let

a. What is the order of

b. If identify and

Solution

a. The matrix has 2 rows and 3 columns, so it is of order

b. The element is in the second row and third column. Thus, The

element is in the first row and second column. Consequently, a12 = 2.a12

a23 = - 15

.a23

2 * 3.

a12 .a23A = 3aij4,

A?

A = B 3 2 0-4 -5 -

15R .

EXAMPLE 1

m = n,nmm : n

Aa32jthithaij .A

A=[aij]Matrix A withelements aij

C,A, B,

S e c t i o n 8.3

� Use matrix notation.

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 827

828 Chapter 8 Matrices and Determinants

Check Point 1 Let

a. What is the order of b. Identify and

Equality of MatricesTwo matrices are equal if and only if they have the same order and correspondingelements are equal.

a31 .a12A?

A = C 5 -2-3 p

1 6S .

� Understand what is meant byequal matrices.

� Add and subtract matrices.

Definition of Equality of MatricesTwo matrices and are equal if and only if they have the same order and for and j = 1, 2, Á , n.i = 1, 2, Á , maij = bij

m * nBA

For example, if and then if and only

if (so ), and

Matrix Addition and SubtractionTable 8.1 shows that matrices of the same order can be added or subtracted by simplyadding or subtracting corresponding elements.

z = 3.y = 4x = 1, y + 1 = 5

A = BB = B1 53 6

R ,A = Bx y + 1z 6

R

Table 8.1 Adding and Subtracting MatricesLet and be matrices of order m * n.B = 3bij4A = 3aij4

Definition The Definition in Words Example

Matrix Addition

A + B = 3aij + bij4

Matrices of the same order are added byadding the elements in correspondingpositions.

= B1 + 1-12 -2 + 63 + 0 5 + 4

R = B0 43 9

RB1 -2

3 5R + B -1 6

0 4R

Matrix Subtraction

A - B = 3aij - bij4

Matrices of the same order are subtracted bysubtracting the elements in correspondingpositions.

= B1 - 1-12 -2 - 63 - 0 5 - 4

R = B2 -83 1

RB1 -2

3 5R - B -1 6

0 4R

The sum or difference of two matrices of different orders is undefined. Forexample, consider the matrices

The order of is the order of is These matrices are of differentorders and cannot be added or subtracted.

3 * 2.B2 * 2;A

A = B0 34 3

R and B = C1 94 52 3

S .

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 828


Adding and Subtracting Matrices

Perform the indicated matrix operations:

a.

b.

Solution

a.

Add the corresponding elementsin the matrices.

Simplify.

b.

Subtract the correspondingelements in the matrices.

Simplify.

Check Point 2 Perform the indicated matrix operations:

a. b.

A matrix whose elements are all equal to 0 is called a zero matrix. If isan matrix and 0 is the zero matrix, then Forexample,

The zero matrix is called the additive identity for matrices.For any matrix the additive inverse of written is the matrix with the

same order as such that every element of is the opposite of the correspondingelement of Because corresponding elements are added in matrix addition,

is a zero matrix. For example,

B -5 23 6

R + B 5 -2-3 -6

R = B0 00 0

R .

A + 1-A2A.

-AA-A,A,A,

m * nm * n

B -5 23 6

R + B0 00 0

R = B -5 23 6

R .

A + 0 = A.m * nm * nA

C 5 4-3 7

0 1S - C -4 8

6 0-5 3

S .B -4 37 -6

R + B6 -32 -4

R

= B -1 12 1

R2 * 2 = B -6 - 1-52 7 - 6

2 - 0 -3 - 1-42R

B -6 72 -3

R - B -5 60 -4

R = B -2 8 8

5 -3 -2R

2 * 3 = B0 + 1-22 5 + 3 3 + 5-2 + 7 6 + 1-92 -8 + 6

R B 0 5 3

-2 6 -8R + B -2 3 5

7 -9 6R

B -6 72 -3

R - B -5 60 -4

R .

B 0 5 3-2 6 -8

R + B -2 3 57 -9 6

REXAMPLE 2Technology

Graphing utilities can add and sub-tract matrices. Enter the matrices andname them and Then use akeystroke sequence similar to

Consult your manual and verify theresults in Example 2.

�3A4� �-� �3B4� �ENTER�

�3A4� �+� �3B4� �ENTER�

3B4.3A4

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 829

TechnologyYou can verify the algebraic solutionin Example 3(b) by first entering thematrices and into yourgraphing utility. The screen belowshows the required computation.

3B43A4

Scalar MultiplicationA matrix of order such as [6], contains only one entry.To distinguish this matrixfrom the number 6, we refer to 6 as a scalar. In general, in our work with matrices, wewill refer to real numbers as scalars.

To multiply a matrix by a scalar we multiply each entry in by For example,

4 =

MatrixScalar

B R2

–35

0B R4(2)

4(–3)4(5)

4(0)= B R .

8

–1220

0

c.Ac,A

1 * 1,


Properties of Matrix AdditionIf and are matrices and 0 is the zero matrix, then thefollowing properties are true.

1. Commutative property of addition

2. Associative property of addition

3. Additive identity property

4. Additive inverse propertyA + 1-A2 = 1-A2 + A = 0

A + 0 = 0 + A = A

1A + B2 + C = A + 1B + C2

A + B = B + A

m * nm * nCA, B,

Definition of Scalar MultiplicationIf is a matrix of order and is a scalar, then the matrix is the

matrix given by

This matrix is obtained by multiplying each element of by the real number Wecall a scalar multiple of A.cA

c.A

cA = 3caij4.

m * ncAcm * nA = 3aij4

� Perform scalar multiplication.

Scalar Multiplication

If and find the following matrices:

a. b.

Solution

b.

= B 4 -121 -18

R

= +

Perform the addition of these 2 x 2matrices by adding corresponding elements.

B R–2

68

0B R6

15–9

–18= B R–2+6

6+158+(–9)

0+(–18)

= +

Multiply eachelement in A by 2.

Multiply eachelement in B by 3.

B R2(–1)

2(3)2(4)

2(0)B R3(2)

3(5)3(–3)

3(–6)

2A + 3B = 2B -1 43 0

R + 3B2 -35 -6

R

a. –5B=–5 =

Multiply each element by −5.

B R2

5–3

–6B R–5(2)

–5(5)–5(–3)

–5(–6)= B R–10

–2515

30

2A + 3B.-5B

B = B2 -35 -6

R ,A = B -1 43 0

REXAMPLE 3

Properties of matrix addition are similar to properties for adding real numbers.

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 830

Check Point 3 If and find the following

matrices:

a. b.

Properties of scalar multiplication are similar to properties for multiplyingreal numbers.

3A + 2B.-6B

B = B -1 -28 5

R ,A = B -4 13 0

RSection 8.3 Matrix Operations and Their Applications 831

Have you noticed the many similarities between addition of real numbers andmatrix addition, subtraction of real numbers and matrix subtraction, andmultiplication of real numbers and scalar multiplication? Example 4 shows howthese similarities can be used to solve matrix equations involving matrix addition,matrix subtraction, and scalar multiplication.

Solving a Matrix Equation

Solve for in the matrix equation

where and

Solution We begin by solving the matrix equation for

This is the given matrix equation.

Subtract matrix from both sides.

12

12

We multiply both sidesby rather than divide X= (B-A)

both sides by 2. This is inanticipation of performing

scalar multiplication.

A 2X = B - A

2X + A = B

X.

B = B -6 59 1

R .A = B1 -50 2

R2X + A = B,

X

EXAMPLE 4

DiscoveryVerify each of the four propertieslisted in the box using

and d = 2.c = 4,

B = B4 01 -6

R ,

A = B 2 -4-5 3

R ,

Properties of Scalar MultiplicationIf and are matrices, and and are scalars, then the followingproperties are true.

1. Associative property of scalar multiplication

2. Scalar identity property

3. Distributive property

4. Distributive property1c + d2A = cA + dA

c1A + B2 = cA + cB

1A = A

1cd2A = c1dA2

dcm * nBA

Multiply both sides by and solve for matrix X.

12

Now we use the matrices and to find the matrix

X =

12

¢ B -6 59 1

R - B1 -50 2

R ≤X.BA

=

12

B -7 109 -1

R Subtract matrices by subtracting corresponding elements.

= D - 72

5

92

- 12

TPerform the scalar multiplication by multiplying each element by 1

2 .

� Solve matrix equations.

Substitute the matrices intoX =

121B - A2.

Take a few minutes to show that this matrix satisfies the given equation Substitute the matrix for and the given matrices for and into the equation.Thematrices on each side of the equal sign, and should be equal.B,2X + A

BAX2X + A = B.

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 831


Check Point 4 Solve for in the matrix equation where

Matrix MultiplicationWe do not multiply two matrices by multiplying the corresponding entries of thematrices. Instead, we must think of matrix multiplication as row-by-column multipli-cation. To better understand how this works, let’s begin with the definition of matrixmultiplication for matrices of order 2 * 2.

A = B2 -80 4

R and B = B -10 1-9 17

R .

3X + A = B,X

Definition of Matrix Multiplication: Matrices

AB=

Row 1 of A× Column 1

of B


of B


of B


of B

B Ra

cb

dB Re

gf

h=B Rae+bg

ce+dgaf+bh

cf+dh

2 : 2

B Ra b B Re

g

Correspondingelements

Correspondingelements

Figure 8.6 Finding correspondingelements when multiplying matrices

� Multiply matrices.

Notice that we obtain the element in the row and column in byperforming computations with elements in the row of and the columnof For example, we obtain the element in the first row and first column of by performing computations with elements in the first row of and the firstcolumn of

You may wonder how to find the corresponding elements in step 1 in the voiceballoon. The element at the far left of row 1 corresponds to the element at the topof column 1. The second element from the left of row 1 corresponds to the secondelement from the top of column 1. This is illustrated in Figure 8.6.

Multiplying Matrices

Find given

A = B2 34 7

R and B = B0 15 6

R .

AB,

EXAMPLE 5

B Ra b B Re

g=B Rae+bg

First rowof A

First columnof B

1. Multiply each element in row 1 of A by thecorresponding element in column 1 of B.

2. Add these products.

3. Record the sum as the element in row 1,column 1 of the product matrix.

B.A

ABB.jthAith

ABjthith

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 832


Solution We will perform a row-by-column computation.

Check Point 5 Find given and

We can generalize the process of Example 5 to multiply an matrix andan matrix. For the product of two matrices to be defined, the number ofcolumns of the first matrix must equal the number of rows of the second matrix.

n*pSecond Matrix

m*nFirst Matrix

The number of columns in the first matrix must bethe same as the number of rows in the second matrix.

n * pm * n

B = B4 61 0

R .A = B1 32 5

RAB,

=

Row 1 of A × Column 1 of B

Row 2 of A × Column 1 of B Row 2 of A × Column 2 of B

Row 1 of A × Column 2 of B

B R2(0)+3(5)

4(0)+7(5)2(1)+3(6)

4(1)+7(6)=B R15

3520

46

AB = B2 34 7

R B0 15 6

R

Definition of Matrix MultiplicationThe product of an matrix, and an matrix, is an matrix, whose elements are found as follows: The element in the rowand column of is found by multiplying each element in the row of by the corresponding element in the column of and adding the products.Bjth

AithABjthithAB,m * pB,n * pA,m * n

Study TipThe following diagram illustrates thefirst sentence in the box definingmatrix multiplication.The diagram ishelpful in determining the order ofthe product AB.

To find a product each row of must have the same number of elements aseach column of We obtain the element in the row and column in byperforming computations with elements in the row of and the column of

When multiplying corresponding elements, keep in mind that the element at the farleft of row corresponds to the element at the top of column The element secondfrom the left in row corresponds to the element second from the top in column Likewise, the element third from the left in row corresponds to the element thirdfrom the top in column and so on.


Matrices and are defined as follows:

Find each product: a. b. BA.AB

A = 31 2 34 B = C456S .

BA

EXAMPLE 6

j,i

j.ij.i

* n�D DT T*

n

� D Tpij=

ith row of A jth column of BElement in the ith row and

jth column of AB

.

B:jthAithAB,jthithpij ,B.

AAB,Matrix A Matrix B

Thesemust beequal.

The order of ABis m * p.

m * n n * p

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 833

TechnologyThe screens illustrate the solution ofExample 6 using a graphing utility.


Solution

a. Matrix is a matrix and matrix is a matrix. Thus, the productis a matrix.

A = 31 2 34 B = C456S

1 * 1AB3 * 1B1 * 3A

Matrix A Matrix B

These areequal.

The order of ABis 1 * 1.

1 * 3 3 * 1

Matrix B Matrix A

These areequal.

The order of BAis 3 * 3.

3 * 1 1 * 3

We will perform a row-by-column computation.

Multiply elements in row 1 of by corresponding elements in column 1 of and add the products.Perform the multiplications.

Add.

b. Matrix is a matrix and matrix is a matrix. Thus, the productis a matrix.3 * 3BA

1 * 3A3 * 1B

= 3324

= 34 + 10 + 184B

A = 3112142 + 122152 + 1321624

AB = 31 2 34C456S

We perform a row-by-column computation.

Simplify.

In Example 6, did you notice that and are different matrices? For mostmatrices and Because matrix multiplication is not commutative, becareful about the order in which matrices appear when performing this operation.

B, AB Z BA.ABAAB

= C4 8 125 10 156 12 18

S

=

(4)(1)

(5)(1)

(6)(1)

(4)(2)

(5)(2)

(6)(2)

(4)(3)

(5)(3)

(6)(3)

F V

Row 3 of B ×Column 1 of A









BA = C456S31 2 34

A = 31 2 34 B = C456S

P-BLTZMC08_805-872-hr 21-11-2008 13:26 34


Check Point 6 If and find and


Where possible, find each product:

a. b.

Solution

a. The first matrix is a matrix and the second is a matrix.The productwill be a matrix.2 * 4

2 * 42 * 2

B1 2 3 40 2 -1 6

R B4 21 3

R .B4 21 3

R B1 2 3 40 2 -1 6

REXAMPLE 7

BA.ABB = C137S ,A = 32 0 44

Arthur Cayley

First Matrix Second Matrix

These areequal.

The order of the product is 2 * 4.

2 * 2 2 * 4

We perform a row-by-column computation.

b. B1 2 3 40 2 -1 6

R B4 21 3

R = B4 12 10 28

1 8 0 22R

= B4 + 0 8 + 4 12 - 2 16 + 121 + 0 2 + 6 3 - 3 4 + 18

RRow 2 ×Column 1

Row 1 ×Column 1

B= R4(1)+2(0)

1(1)+3(0)

Row 2 ×Column 2

Row 1 ×Column 2

4(2)+2(2)

1(2)+3(2)

Row 2 ×Column 3

Row 1 ×Column 3

4(3)+2(–1)

1(3)+3(–1)

Row 2 ×Column 4

Row 1 ×Column 4

4(4)+2(6)

1(4)+3(6)

B4 21 3

R B1 2 3 40 2 -1 6

R

First matrix2 : 4

Second matrix2 : 2

These numbers must be the sameto multiply the matrices.

The number of columns in the first matrix does not equal the number of rows inthe second matrix.Thus, the product of these two matrices is undefined.

Check Point 7 Where possible, find each product:

a. b. B2 3 -1 60 5 4 1

R B1 30 2

R .B1 30 2

R B2 3 -1 60 5 4 1

R

Matrices were first studied inten-sively by the English mathemati-cian Arthur Cayley (1821–1895).Before reaching the age of 25, hepublished 25 papers, setting apattern of prolific creativity thatlasted throughout his life. Cayleywas a lawyer, painter, moun-taineer, and Cambridge professorwhose greatest invention was thatof matrices and matrix theory.Cayley’s matrix algebra, especiallythe noncommutativity of multipli-cation opened up anew area of mathematics calledabstract algebra.

1AB Z BA2,

The Granger Collection

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 835

DiscoveryVerify the properties listed in the boxusing

and c = 3.

C = B 1 2-1 1

R ,

B = B1 03 2

R ,

A = B 3 2-1 4

R ,


Properties of Matrix MultiplicationIf and are matrices and is a scalar, then the following properties are true.(Assume the order of each matrix is such that all operations in these propertiesare defined.)

1. Associative Property of Matrix Multiplication

2. Distributive Properties of MatrixMultiplication

3. Associative Property of Scalar Multiplicationc1AB2 = 1cA2B

1A + B2C = AC + BC A1B + C2 = AB + AC

1AB2C = A1BC2

cCA, B,

ApplicationsAll of the still images that you see on the Web have been created or manipulated on acomputer in a digital format—made up of hundreds of thousands, or even millions, oftiny squares called pixels. Pixels are created by dividing an image into a grid.The com-puter can change the brightness of every square or pixel in this grid. A digital cameracaptures photos in this digital format.Also, you can scan pictures to convert them intodigital format. Example 8 illustrates the role that matrices play in this new technology.

Matrices and Digital Photography

The letter L in Figure 8.7 is shown using 9 pixels in a grid. The colorspossible in the grid are shown in Figure 8.8. Each color is represented by a specificnumber: 0, 1, 2, or 3.

3 * 3

EXAMPLE 8

� Model applied situations withmatrix operations.

Figure 8.7 The letter L

0 1 2 3

White Light gray Dark gray Black

Figure 8.8 Color levels

a. Find a matrix that represents a digital photograph of this letter L.b. Increase the contrast of the letter L by changing the dark gray to black and the

light gray to white. Use matrix addition to accomplish this.

Solutiona. Look at the L and the background in Figure 8.7. Because the L is dark gray,

color level 2, and the background is light gray, color level 1, a digital photo-graph of Figure 8.7 can be represented by the matrix

b. We can make the L black, color level 3, by increasing each 2 in the above matrixto 3. We can make the background white, color level 0, by decreasing each 1 inthe above matrix to 0.This is accomplished using the following matrix addition:

The picture corresponding to the matrix sum to the right of the equal sign isshown in Figure 8.9.

C2 1 12 1 12 2 1

S + C1 -1 -11 -1 -11 1 -1

S = C3 0 03 0 03 3 0

S .

C2 1 12 1 12 2 1

S .

Figure 8.9 Changingcontrast: the letter L

Although matrix multiplication is not commutative, it does obey many of theproperties of real numbers.

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 836


Check Point 8 Change the contrast of the letter L in Figure 8.7 on the previouspage by making the L light gray and the background black. Use matrix addition toaccomplish this.

Images of space

We have seen how functions can be transformed using translations, reflections,stretching, and shrinking. In a similar way, matrix operations are used to transformand manipulate computer graphics.

Transformations of an Image

The quadrilateral in Figure 8.10 can be represented by the matrix

Each column in the matrix gives the coordinates of a vertex, or corner, of thequadrilateral. Use matrix operations to perfom the following transformations:

a. Move the quadrilateral 4 units to the right and 1 unit down.

b. Shrink the quadrilateral to half its perimeter.

c. Let Find What effect does this have on the quadrilateral

in Figure 8.10?

Solution

a. We translate the quadrilateral 4 units right and 1 unit down by adding 4 to eachand subtracting 1 from each This is accomplished

using the following matrix addition:

Each column in the matrix on the right gives the coordinates of a vertex of thetranslated quadrilateral. The original quadrilateral and the translated imageare shown in Figure 8.11.

This matrix represents theoriginal quadrilateral.

Shift 4 units to the rightand 1 unit down.

This matrix represents thetranslated quadrilateral.

B R–2

–3

–1

2

1

–2

3

4+B R4

–1

4

–1

4

–1

4

–1=B R2

–4

3

1

5

–3 .

7

3

y-coordinate.x-coordinate

BA.B = B -1 00 1

R .

Coordinates of vertices

x-coordinates

y-coordinatesB R–2

–3

–1

2

1

–2

3

4A= .

EXAMPLE 9

(3, 4)

(1, −2)

x

y

1 2 3 4 5−1

12345

−2−3−4−5

−1−2−3−4−5

(−2, −3)

(−1, 2)

Figure 8.10

(7, 3)

(2, −4)(5, −3)

(3, 1)

(3, 4)

76x

y

1 2 3 4 5−1

12345

−2−3−4−5

−1−2−3−4−5

(−1, 2)

(1, −2)

(−2, −3)

Figure 8.11 Shifting the quadrilateral4 units right and 1 unit down

Photographs sent back from space use matrices withthousands of pixels. Each pixel is assigned a numberfrom 0 to 63 representing its color–0 for pure white and63 for pure black. In the image of Saturn shown here,matrix operations provide false colors that emphasizethe banding of the planet’s upper atmosphere.

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 837


b. We shrink the quadrilateral in Figure 8.10, shown in blue in Figure 8.12, to halfits perimeter by multiplying each and each by This is accomplished using the following scalar multiplication:

Each column in the matrix on the right gives the coordinates of a vertex of thereduced quadrilateral. The original quadrilateral and the reduced image areshown in Figure 8.12.

c. We begin by finding Keep in mind that represents the originalquadrilateral, shown in blue in Figure 8.13.

ABA.

B =R–2

–3

–1

2

1

–2

3

4B R–1

1 –1 .212 –

–32

12

32

12

This matrix represents theoriginal quadrilateral.

This matrix represents the quadrilateralwith half the original perimeter.

12 .y-coordinatex-coordinate

(−3, 4)

(−2, −3)

(3, 4)

(1, 2)(−1, 2)

(2, −3)

(1, −2)(−1, −2)

x

y

1 2 3 4 5−1

12345

−2−3−4−5

−1−2−3−4−5

Figure 8.13

(1, −2)

(3, 4)

(−2, −3)

(−1, 2)

x

y

1 2 3 4 5−1

12345

−2−3−4−5

−1−2−3−4−5

(w, 2)

(q, −1)

(−1, −w)

(−q, 1)

Figure 8.12 Shrinking thequadrilateral to half the originalperimeter

= B 2 1 -3 -1-3 2 4 -2

R = B1-121-22 + 01-32 1-121-12 + 0122 1-12132 + 0142 1-12112 + 01-22

01-22 + 11-32 01-12 + 1122 0132 + 1142 0112 + 11-22R

BA = B -1 00 1

R B -2 -1 3 1-3 2 4 -2

R

Each column in the matrix multiplication gives the coordinates of a vertex ofthe transformed image. The original quadrilateral and this transformedimage are shown in Figure 8.13. Notice that each on the originalblue image is replaced with its opposite on the transformed red image.

We can conclude that multiplication by reflected the blue quadrilateral about the

Check Point 9 Consider the triangle represented by the matrix

Use matrix operations to perform the following transformations:

a. Move the triangle 3 units to the left and 1 unit down.

b. Enlarge the triangle to twice its original perimeter.

Illustrate your results in parts (a) and (b) by showing the original triangle and thetransformed image in a rectangular coordinate system.

c. Let Find What effect does this have on the original

triangle?

BA.B = B1 00 -1

R .

A = B0 3 40 5 2

R .

y-axis.B -1 0

0 1R

x-coordinate

Exercise Set 8.3

Practice ExercisesIn Exercises 1–4,

a. Give the order of each matrix.

b. If identify and or explain whyidentification is not possible.

1. 2.

3. 4. C -4 1 3 -52 -1 p 01 0 -e 1

5

SC 1 -5 p e

0 7 -6 -p

-2 12 11 -

15

SB -6 4 -1

-9 0 12RB 4 -7 5

-6 8 -1R

a23 ,a32A = 3aij4,

In Exercises 5–8, find values for the variables so that the matricesin each exercise are equal.

5. 6. Bx

7R = B11

yRBx

4R = B6

yR

7. 8. B x y + 32z 8

R = B12 56 8

RBx 2y

z 9R = B4 12

3 9R

In Exercises 9–16, find the following matrices:

a. b.

c. d. 3A + 2B.-4A

A - BA + B

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 838


9. A = B4 13 2

R , B = B5 90 7

R 32.

33.

34.

35.

36.

In Exercises 37–44, perform the indicated matrix operations giventhat and are defined as follows. If an operation is notdefined, state the reason.

37. 38.

39. 40.

41. 42.

43. 44.

Practice PlusIn Exercises 45–50, let

45. Find the product of the sum of and and the differencebetween and

46. Find the product of the difference between and and thesum of and

47. Use any three of the matrices to verify a distributive property.

48. Use any three of the matrices to verify an associative property.

In Exercises 49–50, suppose that the vertices of a computer graphicare points, represented by the matrix

49. Find and explain why this reflects the graphic about the

50. Find and explain why this reflects the graphic about they-axis.

CZ

x-axis.BZ

Z = Bx

yR .

1x, y2,

D.CBA

D.CBA

D = B -1 00 -1

R .

A = B1 00 1

R , B = B1 00 -1

R , C = B -1 00 1

R ,

A1CB2A1BC2

B - AA - C

A1B + C2BC + CB

5C - 2B4B - 3C

A = C 4 0-3 5

0 1S B = B 5 1

-2 -2R C = B 1 -1

-1 1R

CA, B,

A = B2 -1 3 21 0 -2 1

R , B = D-1 2

1 13 -46 5

T

A = B2 -3 1 -11 1 -2 1

R , B = D1 2

-1 15 4

10 5

T

A = C2 43 14 2

S , B = B 3 2 0-1 -3 5

RA = C4 2

6 13 5

S , B = B 2 3 4-1 -2 0

RA = C1 -1 1

5 0 -23 -2 2

S , B = C1 1 01 -4 53 -1 2

S10.

11.

12.

13.

14.

15.

16.

In Exercises 17–26, let

Solve each matrix equation for

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

In Exercises 27–36, find (if possible) the following matrices:

a. b.

27.

28.

29.

30.

31. A = C 1 -1 44 -1 32 0 -2

S , B = C1 1 01 2 41 -1 3

SA = C -1

-2-3S , B = 31 2 34

A = 31 2 3 44, B = D1234

T

A = B3 -21 5

R , B = B0 05 -6

RA = B1 3

5 3R , B = B 3 -2

-1 6R

BA.AB

4B + 3A = -2X4A + 3B = -2X

A - X = 4BB - X = 4A

2X + 5A = B3X + 2A = B

3X + A = B2X + A = B

X - B = AX - A = B

X.

A = C -3 -72 -95 0

S and B = C -5 -10 03 -4

S .

A = C 6 -3 56 0 -2

-4 2 -1S , B = C -3 5 1

-1 2 -62 0 4

SA = C 2 -10 -2

14 12 104 -2 2

S , B = C 6 10 -20 -12 -4

-5 2 -2S

A = 36 2 -34, B = 34 -2 34

A = C 2-4

1S , B = C -5

3-1S

A = B 3 1 1-1 2 5

R , B = B 2 -3 6-3 1 -4

RA = C 1 3

3 45 6

S , B = C2 -13 -20 1

SA = B -2 3

0 1R , B = B8 1

5 4R

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 839


Application Exercisesin the figure is shown using 9 pixels in a

The color levels are given to the right of the figure. Each color isrepresented by a specific number: 0, 1, 2, or 3. Use this informationto solve Exercises 51–52.

3 * 3 grid.The + sign

55. Reduce the L to half its perimeter and move the reducedimage 1 unit up. Then graph the letter and its transformation.

56. Reduce the L to half its perimeter and move the reducedimage 2 units up.Then graph the letter and its transformation.

57. a. If find

b. Graph the object represented by matrix What effectdoes the matrix multiplication have on the letter L repre-sented by matrix

58. a. If find


59. a. If find


60. a. If find


61. Completing the transition to adulthood is measured by oneor more of the following: leaving home, finishing school, get-ting married, having a child, or being financially indepen-dent. The bar graph shows the percentage of Americans, ages20 and 30, who had completed the transition to adulthood in1960 and in 2000.

B?

AB.

AB.A = B2 00 1

R ,

B?

AB.

AB.A = B0 -11 0

R ,

B?

AB.

AB.A = B -1 00 1

R ,

B?

AB.

AB.A = B1 00 -1

R ,

0 1 2 3

White Light gray Dark gray Black

51. a. Find a matrix that represents a digital photograph of

b. Adjust the contrast by changing the black to dark grayand the light gray to white. Use matrix addition toaccomplish this.

c. Adjust the contrast by changing the black to light grayand the light gray to dark gray. Use matrix addition toaccomplish this.

52. a. Find a matrix that represents a digital photograph of

b. Adjust the contrast by changing the black to dark grayand the light gray to black. Use matrix addition toaccomplish this.

c. Adjust the contrast by leaving the black alone andchanging the light gray to white. Use matrix addition toaccomplish this.

The figure shows the letter in a rectangular coordinate system.

The figure can be represented by the matrix

Each column in the matrix describes a point on the letter. Theorder of the columns shows the direction in which a pencil mustmove to draw the letter. The is completed by connecting the lastpoint in the matrix, (0, 5), to the starting point, (0, 0). Use theseideas to solve Exercises 53–60.

53. Use matrix operations to move the L 2 units to the left and3 units down. Then graph the letter and its transformation ina rectangular coordinate system.

54. Use matrix operations to move the L 2 units to the right and3 units down. Then graph the letter and its transformation ina rectangular coordinate system.

L

B = B0 3 3 1 1 00 0 1 1 5 5

R .

x−1

21

34

−2−3−4−5

1 2 3 4 5−1−2−3−4

y(1, 5)

(1, 1) (3, 1)

(3, 0)

(0, 5)

(0, 0)

−5

L

the + sign.

the + sign.

80%

70%

60%

50%

40%

30%

20%Per

cent

age

Hav

ing

Com

plet

ed th

e Tr

ansi

tion

Percentage Having Completed the Transition to Adulthood

Year1960

Age20

Age30

9%10%

Men Women65%

2000

Age20

Age30

2%

31%

1960

Age20

Age30

29%

77%

2000

Age20

Age30

6%

46%

Source: James M. Henslin, Sociology, Eighth Edition, Allyn and Bacon, 2007

a. Use a matrix to represent the data for 2000.Entries in the matrix should be percents that areorganized as follows:

Call this matrix

b. Use a matrix to represent the data for 1960. Callthis matrix

c. Find What does this matrix represent?B - A.

B.2 * 2

A.

Age 20Age 30

Men WomenB R .

2 * 2

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 840


Midterm FinalSystem

1System

2

Student 1Student 2Student 3Student 4Student 5

E76 9274 8494 8684 6258 80

UMidtermFinal

B0.5 0.30.5 0.7

R

62. The table gives an estimate of basic caloric needs fordifferent age groups and activity levels.

Writing in Mathematics65. What is meant by the order of a matrix? Give an example

with your explanation.

66. What does mean?

67. What are equal matrices?

68. How are matrices added?

69. Describe how to subtract matrices.

70. Describe matrices that cannot be added or subtracted.

71. Describe how to perform scalar multiplication. Provide anexample with your description.

72. Describe how to multiply matrices.

73. Describe when the multiplication of two matrices is not defined.

74. If two matrices can be multiplied, describe how to determinethe order of the product.

75. Low-resolution digital photographs use 262,144 pixels in agrid. If you enlarge a low-resolution digital photo-

graph enough, describe what will happen.

Technology Exercise76. Use the matrix feature of a graphing utility to verify each of

your answers to Exercises 37–44.

Critical Thinking ExercisesMake Sense? In Exercises 77–80, determine whether eachstatement makes sense or does not make sense, and explainyour reasoning.

77. I added matrices of the same order by adding correspondingelements.

78. I multiplied an matrix and an matrix by multi-plying corresponding elements.

79. I’m working with two matrices that can be added but notmultiplied.

80. I’m working with two matrices that can be multiplied but notadded.

81. Find two matrices and such that

82. Consider a square matrix such that each element that is noton the diagonal from upper left to lower right is zero.Experiment with such matrices (call each matrix ) byfinding Then write a sentence or two describing amethod for multiplying this kind of matrix by itself.

83. If then and are said to be anticommutative.

Are and anticommutative?

Group Exercise84. The interesting and useful applications of matrix theory are

nearly unlimited. Applications of matrices range from repre-senting digital photographs to predicting long-range trendsin the stock market. Members of the group should researchan application of matrices that they find intriguing. Thegroup should then present a seminar to the class about thisapplication.

B = B1 00 -1

RA = B0 -11 0

RBAAB = -BA,

AA.A

AB = BA.BA

n * pm * n

512 * 512

aij

19 – 30

31 – 50

51+

2400

2200

2000

2000

1800

1600

AgeRange

Sedentary

2700

2500

2300

2100

2000

1800

ModeratelyActive

3000

2900

2600

2400

2200

2100

Active

Men WomenMen WomenMen Women

Source: USA Today

a. Use a matrix to represent the daily caloric needs,by age and activity level, for men. Call this matrix

b. Use a matrix to represent the daily caloric needs,by age and activity level, for women. Call this matrix

c. Find What does this matrix represent?

63. The final grade in a particular course is determined by gradeson the midterm and final. The grades for five students andthe two grading systems are modeled by the followingmatrices. Call the first matrix and the second B.A

M - W.

W.3 * 3

M.3 * 3

a. Describe the grading system that is represented bymatrix

b. Compute the matrix and assign each of the fivestudents a final course grade first using system 1 andthen using system 2.69.5 - 79.4 = C, 59.5 - 69.4 = D, below 59.5 = F2

189.5 - 100 = A, 79.5 - 89.4 = B,

AB

B.

64. In a certain county, the proportion of voters in each age groupregistered as Republicans, Democrats, or Independents isgiven by the following matrix, which we’ll call A.

Age

18–30 31–50 Over 50

RepublicansDemocratsIndependents

C0.40 0.30 0.700.30 0.60 0.250.30 0.10 0.05

S

Male Female

18–30Age 31–50

Over 50 C 6000 8000

12,000 14,00014,000 16,000

S

The distribution, by age and gender, of this county’s votingpopulation is given by the following matrix, which we’ll call B.

a. Calculate the product

b. How many female Democrats are there?

c. How many male Republicans are there?

AB.

P-BLTZMC08_805-872-hr 21-11-2008 13:26 Page 841

Objectives

� Find the multiplicative inverseof a square matrix.

� Use inverses to solve matrixequations.

� Encode and decodemessages.

Multiplicative Inverses of Matrices and Matrix Equations

In 1939, Britain’s secret servicehired top chess players, mathe-

maticians, and other masters oflogic to break the code used bythe Nazis in communicationsbetween headquarters and troops.The project, which employed over

10,000 people, broke the code less than a year later, providing the Allies with infor-mation about Nazi troop movements throughout World War II.

S e c t i o n 8.4

8 Mid-Chapter Check PointWhat You Know: We learned to use matrices to solvesystems of linear equations. Gaussian elimination requiredsimplifying the augmented matrix to one with 1s down themain diagonal and 0s below the 1s. Gauss-Jordan elimina-tion simplified the augmented matrix to one with 1s downthe main diagonal and 0s above and below each 1. Such amatrix, in reduced row-echelon form, did not require back-substitution to solve the system.We applied Gaussian elimi-nation to systems with no solution, as well as to representthe solution set for systems with infinitely many solutions,including nonsquare systems. We learned how to performoperations with matrices, including matrix addition, matrixsubtraction, scalar multiplication, and matrix multiplication.

In Exercises 1–5, use matrices to find the complete solution to eachsystem of equations, or show that none exists.

1. 2. c 2x + 4y + 5z = 2x + y + 2z = 1

3x + 5y + 7z = 4c x + 2y - 3z = -7

3x - y + 2z = 82x - y + z = 5

C h a p t e r


Preview Exercises

Exercises 85–87 will help you prepare for the material covered inthe next section.

85. Multiply:

After performing the multiplication, describe what happensto the elements in the first matrix.

Ba11 a12

a21 a22R B1 0

0 1R .

86. Use Gauss-Jordan elimination to solve the system:

87. Multiply and write the linear system represented by thefollowing matrix multiplication:

Ca1 b1 c1

a2 b2 c2

a3 b3 c3

S Cx

y

z

S = Cd1

d2

d3

S .

c -x - y - z = 14x + 5y = 0

y - 3z = 0.

This 1941 RCA radiogram shows anencoded message from the Japanesegovernment.

3. 4. dw + x + y + z = 6w - x + 3y + z = -14w + 2x - 3z = 12

2w + 3x + 6y + z = 1

b x - 2y + 2z = -22x + 3y - z = 1

5.

In Exercises 6–10, perform the indicated matrix operations orsolve the matrix equation for given that and are definedas follows. If an operation is not defined, state the reason.

6. 7. 8.

9. 10. 2X - 3C = BA + C

A1BC2A1B + C22C -12 B

A = C 0 2-1 3

1 0S B = B 4 1

-6 -2R C = B -1 0

0 1R

CA, B,X

c 2x - 2y + 2z = 5x - y + z = 2

2x + y - z = 1

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p-bltzmc08 805-872-hr 21-11-2008 13:26 page 827 matrix...

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