# Elementary Row Operations For Matrices

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