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

1. ## 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?

2. A homogeneous system is of the form

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

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

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

3. so could we have something like this

0 -2 +2 | 0
0 -4 +4 | 0
2 -1 -1 | 0

4. Yes, that is true.

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

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.