Examples of these row operations follow.
Row operation 1
becomes
Interchange row 1
and row 3.Row operation 2
becomes
Multiply the numbers
in row 1 by 3.Row operation 3
becomes C
0
4
1
3
8
0
- 5
- 3
7
C S
2
4
1
3
8
0
9
- 3
7
S
C
6
4
1
9
8
0
27
- 3
7
C S
2
4
1
3
8
0
9
- 3
7
S
C
1
4
2
0
8
3
7
- 3
9
C S
2
4
1
3
8
0
9
- 3
7
S
248 CHAPTER 4 Systems of Linear Equations
Matrix Row Operations1. Any two rows of the matrix may be interchanged.
2. The elements of any row may be multiplied by any nonzero real number.
3. Any row may be changed by adding to the elements of the row the product of
a real number and the corresponding elements of another row.
Multiply the numbers in
row 3 by Add them
to the corresponding
numbers in row 1.The third row operation corresponds to the way we eliminated a variable from a
pair of equations in previous sections.
OBJECTIVE 3 Use row operations to solve a system with two equations.
Row operations can be used to rewrite a matrix until it is the matrix of a system
whose solution is easy to find. The goal is a matrix in the form
or
for systems with two and three equations, respectively. Notice that there are 1’s down
the diagonal from upper left to lower right and 0’s below the 1’s. A matrix written this
way is said to be in row echelon form.
Using Row Operations to Solve a System with Two VariablesUse row operations to solve the system.
We start by writing the augmented matrix of the system.
c
1
2
- 3
1
`
1
- 5
d
2 x+ y=- 5
x- 3 y= 1
EXAMPLE 1
C
1
0
0
a
1
0
b
d
1
3
c
e
f
c S
1
0
a
1
`
b
c
d
The following row operationsproduce new matrices that lead to systems having
the same solutions as the original system.
Write the
augmented matrix.