math111_logo Existence

2. Existence of solutions for any right side

For a fixed matrix A, the consistency of the system Ax = b usually depends on the choice of b (see this example). In this part, we answer the following question: When the system Ax = b has solutions for any choice of the right side b?

Example We have studied the following system before

3x1 + x2 - x3 = b1
x1 - x2 + x3 = b2
2x1 + 2x2 + x3 = b3

with a special choice on the right side. By applying the same row operations as before, the augmented matrix

[ 3 1 -1 b1 ]
1 -1 1 b2
2 2 1 b3

becomes

[ 1 -1 1 b2 ]
0 4 -4 b1 - 3b2
0 0 3 - b1 + b2 + b3

Due to the fact that there is no zero row [0 0 0] in the darker part (which is a row echelon form of the coefficient matrix), there is no row of the form [0 0 0 ≠0] for all choice of b1, b2, b3. Therefore the system is consistent for any choice of the right side.

Example In an earlier example, we already know that the system

3x1 + x2 - x3 = b1
x1 - x2 + x3 = b2
2x1 - x2 + x3 = b3

has no solution for the special choice of b1 = 2, b2 = 2, b3 = 6. Thus the system is not consistent for some right side (i.e., it is not true that the system is consistent for any right side).

We study the problem again by looking at the coefficient matrix only. The same row operations as before change the coefficient matrix

[ 3 1 -1 ]
1 -1 1
2 -1 1

to a row echelon form.

[ 1 -1 1 ]
0 1 -1
0 0 0

The same row operations must also change the augmented matrix

[ 3 1 -1 b1 ]
1 -1 1 b2
2 -1 1 b3

to

[ 1 -1 1 b'1 ]
0 1 -1 b'2
0 0 0 b'3

where b'1, b'2, b'3 are some combinations of b1, b2, b3. Now we can choose some b1, b2, b3 so that the last combination b'3 ≠ 0. Then the system has no solution for this choice. Thus we conclude that the system is not always consistent. Again we emphasis the key reason for this conclusion is that the row echelon form of the coefficient matrix contains a zero row [0 0 0].

You may also check out the always consistent problem for this example. From all these examples, it is easy to conclude the criterion for the existence of solutions for any right side.

The system Ax = b has solutions for any right side b
⇔ There is no zero row in the row echelon form of A
⇔ All rows of A are pivot


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