http://www.ck12.org Chapter 8. Systems and Matrices
[1 0
0 1]
,
[1 0 3
0 1 4
]
,
1 2 0 0 6
0 0 1 0 7
0 0 0 1 − 2
0 0 0 0 0
Putting a matrix into reduced row echelon form is a result of performingGauss-Jordan elimination. The process
illustrated in this concept is named after those mathematicians.
Example A
Put the following matrix into reduced row echelon form.[
3 7
2 5
]
Solution: In each step of the solution, only one of the three row operations will be used. Specific shorthand will
be introduced.[
3 7
2 5
] →
→ · 3 →
[ 3 7
6 15
] →
→ − 2 ·I →
[3 7
0 1
]
Note that the·3 in between the first two matrices indicates that the second row is scaled by a factor of 3. The
− 2 ·Ibetween the next two matrices indicates that the second row has two times the first row subtracted from
it. The[ Iis a roman numeral referring to the row number.
3 7
0 1
]→ − 7 II →
→
[3 0
0 1
]→ · 1
3 →
→
[1 0
0 1
]
Row reducing a 2×2 matrix to become the identity matrix is an exercise that illustrates the fact that the rows were
linearly independent.
Example B
Put the following matrix into reduced row echelon form.
2 4 0
0 3 1
1 2 4
Solution:
2 4 0
0 3 1
1 2 4
→
→
→ −I 2 →
2 4 0
0 3 1
0 0 4
→
→
→ ÷ 4 →
2 4 0
0 3 1
0 0 1
→ ÷ 2 →
→ ÷ 3 →
→
1 2 0
0 1^13
0 0 1
Note that in the preceding step, two operations were used. This is acceptable when the operations do not interfere
or interact with each other.
1 2 0
0 1^13
0 0 1
→
→ −III 3 →
→
1 2 0
0 1 0
0 0 1
→ − 2 II →
→
→
1 0 0
0 1 0
0 0 1
Again, row reducing a 3×3 matrix to become the identity matrix is just an exercise that illustrates the fact that the
rows were linearly independent.
Example C
In a single 3×3 matrix, describe the general approach of Gauss-Jordan elimination. In other words, which locations
would you try to focus on first?
Solution: One approach is to try to get a one in the A position. Then get a zero in position B and position C by
multiplying by a multiple of row 1. Then try to get a zero in position D.