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Math Help - Matrices, Equivalent Conditions

  1. #1
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    Matrices, Equivalent Conditions

    My problem is this:
    State whether the following is true or false, provide a reason.
    1. Ax = O has only the trivial solution if and only if Ax = b has a unique solution for every n x 1 column matrix b.

    I am leaning towards false. The reason for this lies with a list of equivalent conditions in my book:
    If A in an n x n matrix, then the following statements are equivalent:
    1. A is invertible
    2. Ax = b has a unique solution for every n x 1 column matrix b
    3. Ax = O has only the trivial solution
    4. A is row equivalent to  I_n
    5. A can be written as a product of elementary matrices.

    The reason I am leaning towards false is because the problem makes it seem that part one of it (Ax = O ...) can only be true if part two (Ax = b...) is. Perhaps I am misinterpreting however. Thanks for input.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Alterah View Post

    My problem is this:
    State whether the following is true or false, provide a reason.
    1. Ax = O has only the trivial solution if and only if Ax = b has a unique solution for every n x 1 column matrix b.
    it's true because A\bold{x} = \bold{o} has only the trivial solution \bold{x}=\bold{0} if and only if A is invertible and so \bold{x}=A^{-1}\bold{b} is the unique solution of A\bold{x}=\bold{b}.
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