# how is this a homogenneous system w/ unl. many solutions

• Apr 27th 2010, 11:32 AM
thekrown
how is this a homogenneous system w/ unl. many solutions
8. a)

give an example of a 3x3 homogeneous system of linear equations AX=0 which has infinitely many solutions.

how is

[000|0]
[000|0]
[000|0]

a system like this?

my teacher told me this is a cheesy answer to give if it were to appear on a final exam, but is correct.

how so? what do they mean exactly by a homogeneous system with infinitely many solutions?
• Apr 27th 2010, 03:52 PM
Prove It
A homogeneous system is of the form

$A\mathbf{x} = \mathbf{0}$.

To have a solution, $|A| \neq 0$.

To have infinitely many solutions, at least one of the equations is a scalar multiple of another.
• Apr 28th 2010, 04:03 AM
thekrown
so could we have something like this

0 -2 +2 | 0
0 -4 +4 | 0
2 -1 -1 | 0
• Apr 28th 2010, 04:31 AM
HallsofIvy
Yes, that is true.

I notice that Prove It said: "To have a solution, $|A|\ne 0$.

To have infinitely many solutions, at least one of the equations is a scalar multiple of another. "

To have a unique solution the determinant must not be 0. Since a homogeneous system of equations always has the trivial solution (all unknowns equal to 0), a homogenous system of equations has an infinite number of solutions if and only if its determinant is equal to 0.