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Math Help - Linear independence & inversion question

  1. #1
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    Linear independence & inversion question

    "Show that if ad - bc = 0, then the equation Ax = 0 has more than one solution. Why does this imply that A is not invertible?"

    The problem then gives a hint about considering when a=b=0. That's pretty straightforward--the matrix becomes linearly dependent. But what to do after that? What specifically does this question want?
    Last edited by phys251; August 24th 2012 at 06:28 PM. Reason: thx, Deveno.
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  2. #2
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    Re: Linear independence & inversion question

    linearly DEpendent.

    we always know that (0,0) is a solution. so if we have any OTHER solution, A cannot be invertible.

    well, if a = b = 0, then if c = 0, we have the non-zero solution (1,0), and if d = 0, we have the non-zero solution (0,1).

    otherwise, we have the non-zero solution (1,-c/d).

    so suppose a is non-zero. then we have the non-zero solution (-b/a,1).

    why does this show A is non-invertible?

    if we have Ax = 0, with x0 then what do we choose for the value A-1(0)?
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  3. #3
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    Re: Linear independence & inversion question

    In general if Ax= b (not necessarily equal to 0) and A has an inverse then it follows that x= A^{-1}b so the equation has only that one solution. Any time there is more than one solution (or no solution) A must have no inverse.
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