Chapter Ten — Applications of Matrix Algebra

The Humongous Book of Algebra Problems

211

Note: Problems 10.6–10.7 refer to matrix B deﬁned below.

10.7 Verify that B · D = B, assuming that D is the identity matrix identiﬁed in

Problem 10.6.

Calculate the product of the matrices and verify that it is equal to B.

Matrix Row Operations

Swap rows, add rows, or multiply by a number

10.8 Identify the three elementary matrix row operations.

The three elementary row operations are: exchanging the positions of two rows,

multiplying a row by a nonzero number, and replacing a row by adding it to a

multiple of another row in the matrix.

Note: Problems 10.9–10.12 refer to matrix A deﬁned below.

10.9 Perform the row operation: .

The notation R

i

refers to the ith row of the matrix, so this problem refers to the

second row of A (R

2

), and the third row of A (R

3

). The symbol “ ” indicates

that the rows should be switched. To perform the row operation , move

the elements of the second row to the third row and vice versa.

For example, you

could swap the rst

two rows so that the

second row becomes

the rst and vice

versa.

This is the

trickiest of the

three row operations.

To see it in action,

look at Problems

10.11 and 10.12.

Chapter Ten — Applications of Matrix Algebra

The Humongous Book of Algebra Problems

212

Note: Problems 10.9–10.12 refer to matrix A deﬁned below.

10.10 Perform the row operation: .

Multiply each element of the ﬁrst row by 4.

Note: Problems 10.9–10.12 refer to matrix A deﬁned below.

10.11 Perform the row operation: .

Add the corresponding elements of rows one and two together.

Replace the second row with those values.

Notice that the row operation did not affect the elements in the

ﬁrst or third row of A.

Go back

to the original

matrix A, not the

matrix you end up

with in Problem

10.9.

Even though

you used the

numbers in the rst row

to get the new values for

the second row, the rst

row is unaffected by

the row operation when

everything’s all said

and done.

Chapter Ten — Applications of Matrix Algebra

The Humongous Book of Algebra Problems

213

Note: Problems 10.9–10.12 refer to matrix A deﬁned below.

10.12 Perform the row operation: .

Multiply the elements of the third row by –5 and add the results to the elements

of the ﬁrst row.

Replace the ﬁrst row with these values.

Note: In Problems 10.13–10.15, manipulate matrix B, as deﬁned below, using the indicated

elementary row operation in order to make b

11

= 1.

10.13 Switch two rows of the matrix.

Switch the ﬁrst and second rows of the matrix to move element 1

from position b

21

in the matrix to position b

11

.

In Problems

10.13–10.15, you

have the same goal:

adjust the matrix so

that the upper-left

element is 1. You’ll just

reach that goal three

different ways, each

time using a different

row operation.

Chapter Ten — Applications of Matrix Algebra

The Humongous Book of Algebra Problems

214

Note: In Problems 10.13–10.15, manipulate matrix B, as deﬁned below, using the indicated

elementary row operation in order to make b

11

= 1.

10.14 Multiply a row by a nonzero number.

Multiply each element of the ﬁrst row by (the reciprocal of b

11

= –3).

Note: In Problems 10.13–10.15, manipulate matrix B, as deﬁned below, using the indicated

elementary row operation in order to make b

11

= 1.

10.15 Replace a row with the sum of two rows.

Notice that the sum of elements b

11

and b

31

is 1. Apply the row operation

.

The multi-

plicative inverse

property (see Problem

1.41) says that the

product of any real

number and its

reciprocal is 1.

This row

operation

also works:

.

Basically the whole

point of Problems

10.13–10.15 is to show

you that there are a

lot of different ways to

change numbers in a

matrix. The matrices

you end up with, in

each case, look

different.

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