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Math Help - how is this a homogenneous system w/ unl. many solutions

  1. #1
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    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?
    Last edited by thekrown; April 27th 2010 at 11:45 AM.
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  2. #2
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    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.
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  3. #3
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    so could we have something like this

    0 -2 +2 | 0
    0 -4 +4 | 0
    2 -1 -1 | 0
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  4. #4
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    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.
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