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Math Help - Some basic matrix proofs

  1. #1
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    Some basic matrix proofs

    Can anyone help me solve these? I'm totally stuck...

    1. Let A be invertible. Prove that the transpose of A (denoted A^t)
    is invertible and (A^t)^{-1}=(A^{-1})^t

    2. Prove that if A is invertible and AB=0, then B=0.

    Any help on these?
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  2. #2
    Math Engineering Student
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    \left( A^{-1} \right)^{t}\cdot A^{t}=\left( A\cdot A^{-1} \right)^{t}=I_{n}.

    Thus \left( A^{-1} \right)^{t}\cdot A^{t}\cdot \left( A^{t} \right)^{-1}=I_{n}\cdot \left( A^{t} \right)^{-1}\implies \left( A^{-1} \right)^{t}=\left( A^{t} \right)^{-1}.\quad\blacksquare
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  3. #3
    Super Member Random Variable's Avatar
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     AB = 0

     A^{-1} AB = A^{-1} 0 = 0

     (A^{-1}A)B = 0

    B = 0

    I have no idea why it's making everything lowercase.

    EDIT: I guess it was an issue with Firefox.
    Last edited by Random Variable; June 16th 2009 at 04:27 PM.
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  4. #4
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    You could also use the fact that \det(A)=\det(A^t).
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  5. #5
    Math Engineering Student
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    as far as paupsers knows, since these are basic proofs then the use of determinants wouldn't be valid, though he(she) may clarify this.
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