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Math Help - Matrix Operations

  1. #1
    Junior Member
    Joined
    Sep 2009
    Posts
    27

    Matrix Operations

    Hi all; I have two quick Linear Algebra questions that I am struggling with. I think they are connected.

    #1) Suppose CA = I(sub)n (the n * n identity matrix). Show that the equation Ax = 0 has only the trivial solution. Explain why A cannot have more columns than rows.

    My thoughts: For the second half, I believe this is because if it had more columns than rows, there would be free variables, which would mean that the equation was dependent and not only the trivial solution exists. I can't figure out the proof, however.

    #2) Suppose A is an m * n matrix and there exist n * m matrices C & D such that CA = I(sub)n and AD = I(sub)m. Prove that m = n and C = D. [Hint: Think about the product CAD].

    My thoughts: I don't erally understand the whole In and Im business, for starters. This question has me lost.

    Thanks for the help!!!
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  2. #2
    Junior Member
    Joined
    Sep 2008
    From
    Edmonton
    Posts
    67
    #1
    CA = I
    ==> A is invertible
    ==> Ax = 0 has only the trivial solution
    since Ax = 0
    => A^-1 A x = A^-1 (0) = x
    ==> 0 = x

    if A wasn't invertible then you couldn't make that distinction in general


    #2
    CAD = I_n D = D
    D^-1 CAD = I_m
    D^-1 CAD = D^-1 C (AD) = D^-1 C I_m
    ==> D^-1C = I_m
    ==> C = D
    and you're done
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