Pivot-ing of matrix does not work

[parisse]

(06-25-2024 09:33 PM)jte Wrote:  (Some might teach a Gaussian elimination of this form, but many do not, in introductory linear algebra courses.)

Unfortunately most math teachers ignore the most efficient way to reduce integer matrices by hand. They make a linear combination like this (I assume for simplicity that the matrix is not singular and pivots are on the diagonal):
M[j]=M[j]-M[j][j]/M[pivotline][j]*M[pivotline]
This will introduce fractions, therefore take more time to compute, and it raises the risk to make a computation error.

The more efficient method is called Gauss-Bareiss, it works like this:
M[j]=(M[pivotline][j]*M[j]-M[j][j]*M[pivotline])/(pivot_used_to_reduce_previous_column)
It does not introduce fractions while keeping coefficients not too large.
Addtionally, if you are using reduction to compute a determinant, then run Gauss-Bareiss under the diagonal, once reduced, the determinant is the last diagonal coefficient.