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62 Chapter 2 Rectangular Systems and Echelon Forms
Exercises for section 2.4

2.4.1. Determine the general solution for each of the following homogeneous
systems.

2x + y + z =0,
x 1 +2x 2 + x 3 +2x 4 =0,
4x +2y + z =0,
(a) 2x 1 +4x 2 + x 3 +3x 4 =0, (b)
6x +3y + z =0,
3x 1 +6x 2 + x 3 +4x 4 =0.
8x +4y + z =0.

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

2.4.2. Among all solutions that satisfy the homogeneous system

x +2y + z =0,
2x +4y + z =0,
x +2y − z =0,

determine those that also satisfy the nonlinear constraint y − xy =2z.

2.4.3. Consider a homogeneous system whose coefficient matrix is

12 13 1
 
 24 −13 8 
A =  12 35 7  .


24 26 2
 
36 17 −3
First transform A to an unreduced row echelon form to determine the
general solution of the associated homogeneous system. Then reduce A
to E A , and show that the same general solution is produced.

2.4.4. If A is the coefficient matrix for a homogeneous system consisting of
four equations in eight unknowns and if there are five free variables,
what is rank (A)?