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2.3 Consistency of Linear Systems 55

Solution: Apply Gaussian elimination to the augmented matrix [A|b] as shown:

 1   1 
1221 1 1 221 1
 2 2443 1   0 001 −1 
0
2 2442 2 0 0 000 0
  −→  
3 5865 3 0 2 202 0
1 2 2 1 1
1
 
 0 20 2 0 
2
0 0 0 0 −1
−→   .
1
0 0 0 0 0 0
Because a row of the form ( 0 0 ··· 0 | α ) with α
= 0 never emerges,
the system is consistent. We might also observe that b is a nonbasic column
in [A|b] so that rank[A|b]= rank (A). Finally, by completely reducing A to
E A , it is possible to verify that b is indeed a combination of the basic columns
{A ∗1 , A ∗2 , A ∗5 }.
Exercises for section 2.3

2.3.1. Determine which of the following systems are consistent.

x +2y + z =2, 2x +2y +4z =0,
(a) 2x +4y =2, (b) 3x +2y +5z =0,
3x +6y + z =4. 4x +2y +6z =0.

x − y + z =1, x − y + z =1,
x − y − z =2, x − y − z =2,
(c) (d)
x + y − z =3, x + y − z =3,
x + y + z =4. x + y + z =2.

2w + x +3y +5z =1, 2w + x +3y +5z =7,
4w + 4y +8z =0, 4w + 4y +8z =8,
(e) (f)
w + x +2y +3z =0, w + x +2y +3z =5,
x + y + z =0. x + y + z =3.

2.3.2. Construct a 3 × 4 matrix A and 3 × 1 columns b and c such that
[A|b] is the augmented matrix for an inconsistent system, but [A|c]is
the augmented matrix for a consistent system.

2.3.3. If A is an m × n matrix with rank (A)= m, explain why the system
[A|b] must be consistent for every right-hand side b .