Elementary Row Operations for Matrices
1
0
-3
1
1
0
-3
1
0
8
16
0
2 R2 R2
0
16
32
0
-4
14
2
6
-4
14
2
6
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 (R1) →
0
8
16
0
Row 2 (R2) →
1
0
-3
1
Row 3 (R3) →
-4
14
2
6
↑ ↑ ↑ ↑
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
0
1
0
6
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
Rows can be moved around by switching any two. In this case, R1 and R2 have been
switched.
0
8
16
0
1
0
-3
1
1
0
-3
1
R1 ↔ R2
0
8
16
0
-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, R2 is multiplied by 2.
* Place a 0 in the matrix if
the coefficient of a
variable is 0.
* Make sure only ones are on the diagonal with
0's every other position except for the last
column.
