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218 CHAPTER 7 Matrices and Linear Systems

EXAMPLE 7.21
We will solve the system
2x 1 − 4x 2 + x 3 + x 4 + 6x 5 + 4x 6 − 2x 7 = 0
−4x 1 + x 2 + 6x 3 + 3x 4 + 10x 5 − 3x 6 + 6x 7 = 0
3x 1 + x 2 − 4x 3 + 2x 4 + 5x 5 + x 6 + 3x 7 = 0.

The coefficient matrix is
⎛ ⎞
2 −4 1 1 6 4 −2
A = −4 1 6 3 10 −3 6 ⎠ .
⎝
3 1 −42 5 1 3
We find the reduced matrix
⎛ ⎞
100 3 67/7 4/7 29/7
A R = 010 9/5 178/35 −5/7 118/35 ⎠ .
⎝
00111/5 36/5 0 16/5
7
Since m = 7 and A R has three nonzeros, the solution space is a four-dimensional subspace of R .
The general solution depends on the arbitrary free variables x 4 ,··· , x 7 .Let x 4 =α, x 5 =β, x 6 =γ
and x 7 = δ to write the general solution
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
−3 −67/7 −4/7 −29/7
⎜ −9/5 ⎟ ⎜ −178/35 ⎟ ⎜ 5/7 ⎟ ⎜ −118/35 ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ −11/5 ⎟ ⎜ −36/5 ⎟ ⎜ 0 ⎟ ⎜ −16/5 ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
X = α ⎜ 1 ⎟ + β ⎜ 0 ⎟ + γ ⎜ 0 ⎟ + δ ⎜ 0 ⎟ .
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
0
⎜ ⎟ ⎜ 1 ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
0 0
⎝ ⎠ ⎝ ⎠ ⎝ 1 ⎠ ⎝ 0 ⎠
0 0 0 1
As Example 7.21 suggests, with a little practice, the general solution can be read directly
from the reduced matrix.
A homogenous system always has at least the trivial solution, and may or may not have
nontrivial solutions. Here is a simple condition for a homogeneous system to have a nontrivial
solution.

COROLLARY 7.3

Let A be n × m. Then the homogeneous system AX = O has a nontrivial solution if and only
m − number of nonzero rows of (A R )> 0.

The reason for this is that the system can have a nontrivial solution only when the dimension
of the solution space is positive, having something in it other than the zero vector. Since this
solution space has dimension m − rank(A), there will be a nontrivial solution exactly when this
number is positive.
In particular, look at the case that the system has more equations than unknowns, so m < n.
Since the rank of A cannot exceed the number of rows (equations), in this case
rank(A) ≤ n < m

so m − rank(A)> 0 and the system has a nontrivial solution.

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October 14, 2010 14:23 THM/NEIL Page-218 27410_07_ch07_p187-246