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Elementary Row Operations for Matrices

A. Introduction

A matrix is a rectangular array of numbers - in other words, numbers grouped into rows and

columns. We use matrices to represent and solve systems of linear equations. For example, the

system of equations

8y + 16z = 0

*Make sure to line up all variables and

x

- 3z = 1

leave space if one is missing.

-4x + 14y + 2z = 6

can be represented by what is called an augmented matrix as seen below:

) →

Row 1 (R

0

8

16

0

1

* Place a 0 in the matrix if

) →

the coefficient of a

Row 2 (R

1

0

-3

1

2

variable is 0.

) →

Row 3 (R

-4

14

2

6

3

↑

↑

↑

↑

x

y

z

constant

Coefficients of the three unknown variables ( x, y, and z ) and the constant terms are placed in

their respective places in the matrix.

Solving a system of equations using a matrix means using row operations to get the matrix into the

form called reduced row echelon form like the example below:

1

0

0

3

* Make sure only ones are on the diagonal with

0's every other position except for the last

0

1

0

6

column.

0

0

1

2

This column can have any numbers.

B. Row Operations

We can perform elementary row operations on a matrix to solve the system of linear equations it

represents. There are three types of row operations.

1) Interchanging two rows

R

ows can be moved around by switching any two. In this case, R

and R

have been

1

2

switched.

0

8

16

0

1

0

-3

1

↔ R

1

0

-3

1

R

0

8

16

0

1

2

-4

14

2

6

-4

14

2

6

2) Multiplying a row by a nonzero constant

We can multiply any row by any number except 0. When a row is multiplied by a number,

every element in that row must be multiplied by the same number. Below, R

is multiplied by 2.

2

1

0

-3

1

1

0

-3

1

R

0

8

16

0

0

16

32

0

2 R

2

2

-4

14

2

6

-4

14

2

6

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Parent category: Education