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Proof on invertible matrices

  1. Sep 21, 2008 #1
    1. The problem statement, all variables and given/known data
    Show that the nxn matrix A is invertible iff its determinant is non-zero.
    I think I can do this, but would like the validity checked.

    2. Relevant equations
    I would use |A| = the product of diagonal entries, because I don't know how to prove the non-diagonal entries of zero for [tex]A{C^T} =|A|I[/tex], where C is the cofactor matrix.(shown on Gil's videos)

    3. The attempt at a solution
    Using eliminationto show A in rref, That is, all non diagonal entries of zero, then the det is the product of diagonals. If det=0 then at least one entry of diagonals (a11,a22,...,ann) = 0. Then this 0('s) entry cannot be multiplied to produce the entry in the identity matrix needed for [tex]AA^{-1}=I[/tex]
    Is this ok?, if not , some direction would be most appreciated.
    Last edited: Sep 21, 2008
  2. jcsd
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