Example:
2x + y - 3z = 4
x + 2y - z = 10
3x - 2y + 2z = 5
| 2 1 -3 4|
| 1 2 -1 10|
| 3 -2 2 5|
The first three columns is the matrix of the coefficients of x, y, and z. The last column is the right side of the system of equations.
Elementary Row Operations:
1. Interchange two rows.
2. Multiply a row by a nonzero constant.
3. Add a multiple of a row to another row.
| 2 1 -3 4|
| 1 2 -1 10|
| 3 -2 2 5|
Work column by column.
| 1 1/2 -3/2 2|
| 1 2
-1
10|
| 3 -2
2 5|
Now, put zeros in the two positions below the 1 by multiplying Row 1 by something and adding it to Row 2 or Row 3.
Multiply Row 1 by -1 and add it to Row 2 and put the answer in Row 2.
| 1 1/2
-3/2
2|
| 0 3/2
1/2
0|
| 3
-2
2 5|
Multiply Row 1 by -3 and add it to Row 3 and put the answer in Row 3.
| 1 1/2
-3/2
2|
| 0 3/2
1/2
0|
| 0 -7/2 13/2 -1|
| 1 1/2
-3/2
2|
| 0
1
1/3 16/3|
| 0 -7/2
13/2
-1|
And, now, put 0's in the first and third rows above and below that 1 by taking appropriate actions.
Multiply Row 2 by -1/2 , add it to Row 1, and put the answer in Row 1.
| 1
0
-5/3 -2/3|
| 0
1
1/3 16/3|
| 0 -7/2
13/2
-1|
Multiply Row 2 by 7/2, add it to Row 3, and put the answer in Row 3.
| 1
0
-5/3 -2/3|
| 0
1
1/3 16/3|
| 0
0
23/3 53/3|
Multiply Row 3 by 3/23.
| 1
0
-5/3 -2/3|
| 0
1
1/3 16/3|
| 0
0
1 53/23|
The last thing to do is to make 0's appear in the first and second rows above the 1 in the third row.
Multiply Row 3 by 5/3 and add it to Row 1, putting the answer in Row 1.
| 1
0
0 73/23|
| 0
1
1/3 16/3|
| 0
0
1 53/23|
Multiply Row 3 by -1/3, add it to Row 2, and put the answer in Row 2.
| 1
0
0 73/23|
| 0
1
0 105/23|
| 0
0
1 53/23|
This is in row-reduced form and essentially says x =
73/23,
y = 105/23, and z = 53/23.
Problems (there are no problems here yet)