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Math Help - Invertible matrix proof

  1. #1
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    Invertible matrix proof

    Hello,

    Problem statement: show that if Ax = b has a unique solution, then A is invertible.

    I have a hint given to me also: suppose Ax = b has a solution and consider that Ay = 0. Deduce y = 0.

    I'm not sure how to incorporate this hint into the proof. If someone can help me out, that would be really great. I haven't done mathematical proofs in years while I was working, and now I'm back at school. Any general words of advice on how to start/set up a proof will be greatly appreciated, too! Thanks!
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  2. #2
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    If A is invertible, then the system, Ax=b has only one solution.

    Namely, that solution is x=A^{-1}b.

    Since A(A^{-1}b)=b, then it follows that

    x=A^{-1}b is a solution to Ax=b.

    Assume that x_{0} is an arbitrary solution and then show that

    x_{0} must be a solution to A^{-1}b.

    If x_{0} is any solution, then Ax_{0}=b.

    Multiply both sides by A^{-1}, and we get:

    x_{0}=A^{-1}b
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  3. #3
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    Hey Galactus, I need to assume that there is a unique solution first and prove that A is invertible. I think in your proof you assume A is invertible and showed that there is one solution. Could you give it another try and solve the other way using the hint, "suppose Ax = b has a solution and consider that Ay = 0. Deduce y = 0?"

    Or.. I might not be seeing something here. Thanks regardless.
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  4. #4
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    While it's not the only way to prove it, I like to prove by contradiction. So assume A is not invertible. Then that must mean that A is not 1:1, which means that its kernel contains a non-zero element.

    This is where your hint comes in. If Ay=0, what is A(x+y)?
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