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Matrix invertibility question

  1. Mar 8, 2008 #1
    Suppose A : n x n
    A is invertible iff the columns (and rows) of A are linearly independent

    A is invertible
    iff det |A| is non-zero
    iff rank A is n
    iff column rank is n
    iff dim (column space is n)
    iff the n columns of A are linearly independent

    Well, this is a proof that I laid down. It was junked by my prof. She said I have to use linear transformations to prove it. Someone throw some light ?
  2. jcsd
  3. Mar 12, 2008 #2
    I would suggest looking up umm...other...methods of proof. Such as when you see an "iff", you might want to try and prove that if A is invertible, then the columns and rows of A are linearly independent. Then try to prove that if the rows and columns within A are linearly independent, then A is invertible.

    Also, sometimes questions want you to prove it a certain way...so while you might be able to prove it another way, using other methods or something..it's not necessarily relevent to the topic of discussion...I really don't know since I'm not in your class. Hope this helps.
  4. Mar 18, 2008 #3
    in order to prove matrix invertibility
    thereare two ways
    the ony is to make a row reduction on this matrix
    and if in the end of the process you dont have a line of zeros
    then its invertable.

    the other way is if the determinant of this metrix differs zero then its invertable
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