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Math Help - Linear Algebra Proofs

  1. #1
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    Linear Algebra Proofs

    Prove or give a counter example of the following

    A)
    The set of all n x n invertible matrices is a subspace

    B)
    The set of all n x n non invertible is a subspace.

    C)
    Ifm= n then the row space of A is equal to the colomn space

    D) If A and AT have the same number of pivots

    I'm new here so thanks
    Last edited by tenrear3; October 14th 2012 at 10:59 AM.
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  2. #2
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    Re: Linear Algebra Proofs

    A) if A is invertible, then -A is also invertible (with inverse -A-1), but their sum is the 0-matrix, which is not invertible. this means we do NOT have closure under vector addition of invertible matrices.

    B) consider the matrices:

    A = \begin{bmatrix}1&0\\0&0 \end{bmatrix}; B =  \begin{bmatrix}0&0\\0&1 \end{bmatrix}. what is A+B?

    C) nope. they do have the same dimension, though (look at 3x3 matrices of rank 2).

    D)If A and AT have the same number of pivots...then...what? something is missing, here....
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  3. #3
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    Re: Linear Algebra Proofs

    D) Does A and A*T have the same number of Pivots I think. Any other answers to these problems to expand what the person above said.
    Last edited by tenrear3; October 14th 2012 at 12:40 PM.
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