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Math Help - Need Explanations For True/False Questions

  1. #1
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    Need Explanations For True/False Questions

    I know the following answers are correct, I just need to know why they are correct. If someone could give me a brief explanation as to why. It will greatly help me in understand them. TIA!!!!

    1. The equation A(nxm)X = b has a unique solution for all b contained in R^n if and only if the rank of A is n. (A is an n x m matrix.) False. But why?

    2. Ax = 0 has more than one solution if and only if the columns fo A form a linearly depenedent set. True, but why?

    3. If Ax = b has a general solution u + x2v + x4w then AB = 0 where B is the matrix with columns v and w. (x2 and x4 are x sub 2 and x sub 4) True, but why??

    4. If A and B are invertible n x n matrices, then A + B is invertible. False, but why?????
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  2. #2
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    Quote Originally Posted by TreeMoney View Post

    4. If A and B are invertible n x n matrices, then A + B is invertible. False, but why?????
    Consider,
    A=\left[ \begin{array}{cc}1&0\\0&1 \end{array} \right]
    B=\left[ \begin{array}{cc}-1&0\\0&1 \end{array} \right]
    Note that,
    \det (A), \det (B)\not = 0
    Thus, A,B are intertible.
    But,
    A+B=\left[ \begin{array}{cc}0&0\\0&0 \end{array} \right]
    With,
    \det (A+B)=0
    Thus it is not invertible.
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