Page 239 - Advanced engineering mathematics

7.5 Homogeneous Systems 219

COROLLARY 7.4

A linear homogeneous system with more unknowns than equations always has a nontrivial
solution.

Corollary 7.3 implies that AX = O has only the trivial solution exactly when m minus the
number of nonzero rows of the reduced matrix is zero. In particular, when A is square, then m =n
and this occurs exactly when the n × n matrix A R has n nonzero rows, which in turn happens
exactly when A R = I n .

COROLLARY 7.5
If A is n × n, then AX = O has only the trivial solution if and only if A R = I n .

EXAMPLE 7.22

We will solve the system
−4x 1 + x 2 − 7x 3 = 0
2x 1 + 9x 2 − 13x 3 = 0
x 1 + x 2 + 10x 3 = 0.
The coefficient matrix is
⎛ ⎞
−4 1 −7
A = ⎝ 2 9 −13 ⎠ .
1 1 10
We find that A R = I 3 . Therefore the system has only the trivial solution. This can also be seen
from the reduced system, which is
x 1 = 0

x 2 = 0
x 3 = 0.

SECTION 7.5 PROBLEMS

In each of Problems 1 through 12, determine the dimen- 4. 4x 1 + x 2 − 3x 3 + x 4 = 0
sion of the solution space and find the general solution of 2x 1 − x 3 = 0
the system by reducing the coefficient matrix. Write the
general solution in terms of one or more column matrices. 5. x 1 − x 2 + 3x 3 − x 4 + 4x 5 = 0
2x 1 − 2x 2 + x 3 + x 4 = 0
1. x 1 + 2x 2 − x 3 + x 4 = 0 x 1 − 2x 3 + x 5 = 0
x 2 − x 3 + x 4 = 0 x 3 + x 4 − x 5 = 0
2. −3x 1 + x 2 − x 3 + x 4 + x 5 = 0
6. 6x 1 − x 2 + x 3 = 0
x 2 + x 3 + 4x 5 = 0
−3x 3 + 2x 4 + x 5 = 0 x 1 − x 4 + 2x 5 = 0
x 1 − 2x 5 = 0
3. −2x 1 + x 2 + 2x 3 = 0
x 1 − x 2 = 0
x 1 + x 2 = 0

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October 14, 2010 14:23 THM/NEIL Page-219 27410_07_ch07_p187-246