Page 77 - Matrix Analysis & Applied Linear Algebra

70 Chapter 2 Rectangular Systems and Echelon Forms

Summary

Let [A|b] be the augmented matrix for a consistent m × n nonhomo-
geneous system in which rank (A)= r.

• Reducing [A|b] to a row echelon form using Gaussian elimination
and then solving for the basic variables in terms of the free variables
leads to the general solution

h n−r .
x = p + x f 1 h 1 + x f 2 h 2 + ··· + x f n−r
range over all possible values, this general
As the free variables x f i
solution generates all possible solutions of the system.
• Column p is a particular solution of the nonhomogeneous system.
• The expression x f 1 h 1 + x f 2 h 2 + ··· + x f n−r h n−r is the general so-
lution of the associated homogeneous system.
• Column p as well as the columns h i are independent of the row
echelon form to which [A|b] is reduced.

• The system possesses a unique solution if and only if any of the
following is true.
rank (A)= n = number of unknowns.
There are no free variables.
The associated homogeneous system possesses only the trivial
solution.

Exercises for section 2.5

2.5.1. Determine the general solution for each of the following nonhomogeneous
systems.
2x + y + z =4,
x 1 +2x 2 + x 3 +2x 4 =3,
4x +2y + z =6,
(a) 2x 1 +4x 2 + x 3 +3x 4 =4, (b)
6x +3y + z =8,
3x 1 +6x 2 + x 3 +4x 4 =5.
8x +4y + z =10.

x 1 + x 2 +2x 3 =1, 2x + y + z =2,
3x 1 +3x 3 +3x 4 =6, 4x +2y + z =5,
(c) (d)
2x 1 + x 2 +3x 3 + x 4 =3, 6x +3y + z =8,
x 1 +2x 2 +3x 3 − x 4 =0. 8x +5y + z =8.