Chapter Ten — Applications of Matrix Algebra

The Humongous Book of Algebra Problems

208

Augmented and Identity Matrices

Extra columns and lots of 0s and 1s

10.1 What is an augmented matrix?

An augmented matrix is created by joining together matrices that contain the

same number of rows. They are most commonly used to represent systems of

equations, whereby the coefﬁcient matrix is joined with a column of constants.

Consider matrices A and B deﬁned below.

The augmented matrix is constructed by including the values of matrix B

as an additional column of matrix A. The new column is usually separated from

the original columns of A by a dotted line.

10.2 Construct the augmented matrix for the below system of equations,

such that C is the coefﬁcient matrix and D is the column of constants.

The coefﬁcient matrix for this system contains the x-coefﬁcients in the ﬁrst

column and the y-coefﬁcients in the second.

The column of constants is a 2 × 1 matrix containing the numbers that appear

right of the equal sign.

Construct the augmented matrix by including the elements of D as the

third column of C.

You can

also indicate an

augmented matrix

using a solid, instead

of a dotted, line in the

matrix name [A|B] and

in the matrix itself.

Not sure what a

coefcient matrix or

a column of constants

is? Check out

Problem 9.38.

When a term

has no explicit

coefcient written

(like x in the equation

x + 9y = 11, the

coefcient is 1. When

there’s no coefcient

but the variable is

negative (like –y in

3x – y = 5), the

coefcient is –1.

Chapter Ten — Applications of Matrix Algebra

The Humongous Book of Algebra Problems

209

Note: Problems 10.3–10.5 refer to matrix A deﬁned below.

10.3 Construct matrix B that serves as the additive identity for matrix A and verify

that A + B = A.

As explained in Problem 1.36, the additive identity for real numbers is 0.

Adding 0 to any real number x does not change the value of x.

x + 0 = 0 + x = x

To add matrix B to matrix A without changing the elements of A, every element

of B must be 0. Therefore, the additive identity for any m × n matrix is the zero

matrix 0

m × n

.

Verify that B is the additive inverse by demonstrating that A + B = A.

Note: Problems 10.3–10.5 refer to matrix A deﬁned below.

10.4 Construct matrix C that serves as the multiplicative identity for matrix A.

The identity matrix I

n

is a square matrix with n rows and n columns that

contains 0s for each of its elements except for those along the diagonal that

begins with the element in the ﬁrst row and the ﬁrst column.

In this problem, matrix A has two columns, so for the product A · C to exist, C

must have two rows. Therefore, the correct identity matrix is C = I

2

.

A zero

matrix is

exactly what

you’d imagine:

a matrix that

contains nothing but

zeros. It has to have

the same dimensions

of A because

you can’t add

matrices unless

they’re the

same size.

An identity

matrix has ones

along the diagonal

that stretches from the

top-left to the bottom-

right corner. All the

other elements in

the matrix are

zeros.

Chapter Ten — Applications of Matrix Algebra

The Humongous Book of Algebra Problems

210

Note: Problems 10.3–10.5 refer to matrix A deﬁned below.

10.5 Verify that A · C = A.

According to Problem 10.4, the multiplicative identity is . Calculate

the product of A and C using the technique described in Problems 9.17–9.19.

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

10.6 Construct matrix D that serves as the multiplicative identity for matrix B.

Matrix B contains three columns, so the identity matrix must contain the same

number of rows: D = I

3

.

I

3

, like the

identity matrix

I

2

from Problem

10.4, has all zero

elements except

along the diagonal

that goes from top-

left to bottom-right.

The elements in

that diagonal

are all ones.

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