I have an interesting (well, interesting to me, because of the application) result for a 2x2 linear system that I would like to expand to the nxn case. Any suggestions welcome. Note: * represents matrix multiplication, diag[X] means the diagonal matrix with the elements of X on the diagonal and zeros elsewhere, |X| is the determinant of X, tr[X] is the trace of X.
Given a invertible 2x2 matrix A with elements a_ij and having 1's on the main diagonal, define X=A^(-1)*{1,1}. Additionally, define J=diag[X]*A.
Claim: |J|=tr[X]-1.
Proof: |J|=|diag[X]|*|A|, so using Cramer's rule for X, we see
|diag[X]|=(1-a_12)(1-a_21)/|A|/|A|.
Therefore
|J|=(1-a_12)(1-a_21)/|A|
=((1-a_12)(1-a_21)+|A|)/|A|-1
=(2-a_12*a_21)/|A|-1=tr[X]-1
I know that in the 3x3 case |J|!=tr[X]-1, but I'd like to know if there is some other relationship between |J| and tr[X]. Thanks.