# Proof on invertible matrices

1. Sep 21, 2008

### dim&dimmer

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 $$A{C^T} =|A|I$$, where C is the cofactor matrix.(shown on Gil's videos)
$$AA^{-1}=I$$

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 $$AA^{-1}=I$$
Is this ok?, if not , some direction would be most appreciated.
DIM

Last edited: Sep 21, 2008