Suppose the augmented matrix of a system Ax = b is simplified, by the row operations, to the following reduced row echelon form.
[ | 1 | 2 | 0 | 0 | 3 | ] |
0 | 0 | 1 | 0 | 4 | ||
0 | 0 | 0 | 1 | 5 |
This means that the simplified system is
x_{1} | + 2x_{2} | = | 3 | ||
x_{3} | = | 4 | |||
x_{4} | = | 5 |
Then it is easy to get the general solution
x_{1} = 3 - 2x_{2}
x_{3} = 4
x_{4} = 5
where x_{2} is arbitrary. For a more general case, if the following is the reduced row echelon form of [A b],
[ | 0 | 1 | 0 | a_{1} | 0 | b_{1} | 0 | c_{1} | ] |
0 | 0 | 1 | a_{2} | 0 | b_{2} | 0 | c_{2} | ||
0 | 0 | 0 | 0 | 1 | b_{3} | 0 | c_{3} | ||
0 | 0 | 0 | 0 | 0 | 0 | 1 | c_{4} |
then the solution of the system Ax = b is
x_{2} | = | c_{1} | - a_{1}x_{4} | - b_{1}x_{6} |
x_{3} | = | c_{2} | - a_{2}x_{4} | - b_{2}x_{6} |
x_{5} | = | c_{3} | - b_{3}x_{6} | |
x_{7} | = | c_{4} |
where x_{1}, _{}x_{4}, and _{}x_{6} are arbitrary.
We note that the coefficients in the general solution are explicitly given in the reduced row echelon form. Therefore the general solution of Ax = b corresponds explicitly to the reduced row echelon form of [A b]. This has the following consequences: