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Math Help - Nontrivial solution

  1. #1
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    Nontrivial solution

    How do I prove / disprove "If a Matrix A is nonsingular, then the homogeneous system Ax=0 has a nontrivial solution?"

    I know that if A is nonsingular, then A^{-1} exists and multiplying both sides of Ax=0 by A^{-1} gives A^{-1}(Ax)=A^{-1}(0) so (AA^{-1})(x)=0 then (I_n)x=0 and x=0. Therefore, the only solution to Ax=0 is x=0 so a nonsingular matrix never has a trivial solution to Ax=0.

    Is this enough to disprove the statement?
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  2. #2
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    Quote Originally Posted by antman View Post
    How do I prove / disprove "If a Matrix A is nonsingular, then the homogeneous system Ax=0 has a nontrivial solution?"

    I know that if A is nonsingular, then A^{-1} exists and multiplying both sides of Ax=0 by A^{-1} gives A^{-1}(Ax)=A^{-1}(0) so ({\color{red}A^{-1}A})(x)=0 then (I_n)x=0 and x=0. Therefore, the only solution to Ax=0 is x=0 so a nonsingular matrix never has a nontrivial solution to Ax=0.

    Is this enough to disprove the statement?
    Just a minor change. One, you're multiplying both sides by A^{-1} from the left so you should stay consistent (even though AA^{-1} = A^{-1}A = I).

    And the second red is probably a typo.
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  3. #3
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    Thank you! That does make more sense and that definitely was a typo.
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