Page 61 - Matrix Analysis & Applied Linear Algebra

54 Chapter 2 Rectangular Systems and Echelon Forms

this is no help in determining the value of any unknown, it is nevertheless a true
statement, so it doesn’t indicate inconsistency in the system.
There are some other ways to characterize the consistency (or inconsistency)
of a system. One of these is to observe that if the last column b in the augmented
matrix [A|b] is a nonbasic column, then no pivot can exist in the last column,
and hence the system is consistent because the situation (2.3.1) cannot occur.
Conversely, if the system is consistent, then the situation (2.3.1) never occurs
during Gaussian elimination and consequently the last column cannot be basic.
In other words, [A|b] is consistent if and only if b is a nonbasic column.
Saying that b is a nonbasic column in [A|b] is equivalent to saying that
all basic columns in [A|b] lie in the coefficient matrix A . Since the number of
basic columns in a matrix is the rank, consistency may also be characterized by
stating that a system is consistent if and only if rank[A|b]= rank (A).
Recall from the previous section the fact that each nonbasic column in [A|b]
must be expressible in terms of the basic columns. Because a consistent system
is characterized by the fact that the right-hand side b is a nonbasic column,
it follows that a system is consistent if and only if the right-hand side b is a
combination of columns from the coefficient matrix A.
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Each of the equivalent ways of saying that a system is consistent is sum-
marized below.

Consistency
Each of the following is equivalent to saying that [A|b] is consistent.

• In row reducing [A|b], a row of the following form never appears:
(0 0 ··· 0 | α ) , where α
=0. (2.3.2)

• b is a nonbasic column in [A|b]. (2.3.3)
• rank[A|b]= rank (A). (2.3.4)
• b is a combination of the basic columns in A. (2.3.5)

Example 2.3.1

Problem: Determine if the following system is consistent:
x 1 + x 2 +2x 3 +2x 4 + x 5 =1,
2x 1 +2x 2 +4x 3 +4x 4 +3x 5 =1,
2x 1 +2x 2 +4x 3 +4x 4 +2x 5 =2,
3x 1 +5x 2 +8x 3 +6x 4 +5x 5 =3.

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Statements P and Q are said to be equivalent when (P implies Q) as well as its converse (Q
implies P) are true statements. This is also the meaning of the phrase “P if and only if Q.”