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Linear algebra proof - inverses

  1. Jul 26, 2011 #1
    1. The problem statement, all variables and given/known data

    Let m < n. Let A be an n × m matrix, and let B be an m × n matrix. Prove that AB /= In .


    2. Relevant equations



    3. The attempt at a solution
    since m<n, the reduced form of matrix B will have free variables.
    I know that if A and B are invertible matrices, AB will be as well but does the converse also hold?
     
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  3. Jul 26, 2011 #2

    micromass

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    Hi miky87! :smile:

    This makes no sense. A matrix can only be invertible when it's square. And these are not square matrices.

    What can you tell us about the rank of A, B, AB and In??
     
  4. Jul 26, 2011 #3

    Ray Vickson

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    The question make perfect sense: he is being asked to show that A*B cannot be the identity matrix (i.e., that a non-square matrix cannot have a left- or right-inverse). This seems closely related to another question in this Forum, and the same advice given there applies here.

    RGV
     
  5. Jul 26, 2011 #4

    micromass

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    A matrix A is invertible if and only if there is a matrix B such that AB=BA=I. This can only be satisfied for square matrices. So calling A and B invertible in his question does not make any sense.
    Left and right inverses do make sense, but I doubt he meant that.
     
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