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Proofs Matrices Invertible

  1. Oct 3, 2006 #1
    i need to be able to prove that an nxn matrix with two identical columns cannot be invertible. I know that if the columns of the matrix are linearly independent then the matrix is invertible. Could some plz give me a hint on how to do this proof because i really don't know where to start. :frown:
     
  2. jcsd
  3. Oct 3, 2006 #2

    Galileo

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    Depending on what has been covered in your class, it may be easiest to work with the nullspace of the matrix A. A is invertible iff the nullspace of A contains only the zero vector.
    Call, the matrix A. If you can find a nonzero vector x such that Ax=0, then you've shown A is not invertible.
     
  4. Oct 3, 2006 #3

    HallsofIvy

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    As Galileo said, it depends on what has been covered. It is relatively easy to show that the determinant of a matrix in which two columns are the same is 0.
     
  5. Oct 3, 2006 #4

    Pythagorean

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    what's it mean when two columns of a matrix are identical?

    Compare this with what is meant by 'linearly independent'.

    Can two identical columns in one matrix be independent?
     
  6. Oct 4, 2006 #5

    HallsofIvy

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    Actually, I've never seen a text that definedf "independent" for the columns of a matrix!

    You can, of course, think of the columns of a matrix as vectors and then determine whether or not those vectors are independent.
     
    Last edited: Oct 4, 2006
  7. Oct 4, 2006 #6

    Office_Shredder

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    hallsofivy, I think the columns being independent refers to them as vectors being linearly independent essentially (it's not proper terminology perhaps, but it does get the point across)
     
  8. Oct 4, 2006 #7

    HallsofIvy

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    Yeh, I went back and edited my post just before I saw this.
     
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