Intermediate Algebra (11th edition)

OBJECTIVES

Solving Systems of Linear Equations by Matrix Methods

4.4

1 Define a matrix.

2 Write the
augmented matrix
of a system.

3 Use row operations
to solve a system
with two equations.

to solve a system
with three
equations.

to solve special
systems.

OBJECTIVE 1 Define a matrix. An ordered array of numbers such as

Columns

Rows Matrix

is called a matrix.The numbers are called elementsof the matrix. Matrices(the plu-

ral of matrix) are named according to the number of rowsand columnsthey contain.

The rows are read horizontally, and the columns are read vertically. This matrix is a

(read “two by three”) matrix, because it has 2 rows and 3 columns. The num-

ber of rows followed by the number of columns gives the dimensionsof the matrix.

≥

8

2

0

5

- 1

1

5

9

- 3

6

- 3

7

c ¥

- 1

1

0

- 2

d

2 * 3

c

2

7

3

1

5

2

d

4.4 Solving Systems of Linear Equations by Matrix Methods

matrix

2 * 2 matrix

4 * 3

A square matrixis a matrix that has the same number of rows as columns. The

matrix above is a square matrix.

FIGURE 10shows how a graphing calculator displays the preceding two matrices.

Consult your owner’s manual for details for using matrices.

In this section, we discuss a matrix method of solving linear systems that is a

structured way of using the elimination method. The advantage of this new method is

that it can be done by a graphing calculator or a computer.

OBJECTIVE 2 Write the augmented matrix of a system.To solve a linear

system using matrices, we begin by writing an augmented matrixof the system. An

augmented matrixhas a vertical bar that separates the columns of the matrix into

two groups. For example, to solve the system

start by writing the augmented matrix

c

1

2

- 3

1

`

1

- 5

d.

2 x+ y=-5,

x- 3 y= 1

2 * 2

FIGURE 10

Augmented matrix

Place the coefficients of the variables to the left of the bar, and the constants to the

right. The bar separates the coefficients from the constants. The matrix is just a

shorthand way of writing the system of equations, so the rows of the augmented

matrix can be treated the same as the equations of a system of equations.

Exchanging the positions of two equations in a system does not change the sys-

tem. Also, multiplying any equation in a system by a nonzero number does not

change the system. Comparable changes to the augmented matrix of a system of

equations produce new matrices that correspond to systems with the same solutions

as the original system.