One can obtain an upper triangular form from the LU decomposition.

Sometimes it is not necessary to have a matrix in RREF, only in REF. So if I am doing a matrix reduction by hand and stop at REF, then I would like Mathematica to be able to also stop at REF, so that I can check my calculations.

In what instances would I stop at REF when doing calculations by hand? 1. Sometimes it is faster for me to use back substitution(to get the solution) than to get a matrix to RREF. 2. Another example is for finding the basis for a column space or row space . 3. To deduce whether a system of equations has 0, 1 or infinitely many solutions

I want to use Mathematica to check calculations that I've done by hand(I don't want Mathematica give me a reduced answer). But if I stop at REF, and Mathematica stops at RREF, then I can't use it to check whether my row reductions are correct(without me doing further unnecessary calculations). Much of my coursework involves matrices in REF. Maple has the option to get a matrix to REF and/or RREF, and I found this feature very useful.

As a new user of Mathematica, the solution in your first reply, while it did generate a REF, it is not a trivial solution for a beginner . I don't even know what "LU decomposition" is. If your solution can be packaged as a clickable function in Mathematica, then that would be much easier and quicker to implement.

Anyway, that's all the time I want to devote to this topic. I think that I can probably explain myself a bit better, but I have neither the time or the inclination. Thanks for your replies. :)

Thanks a lot for your reply, Daniel.

It would be great if Mathematica had a Gaussian elimination function, as well as a Gauss-Jordan elimination function. Maple has both of these options.

Gaussian elimination stops once the matrix is in row echelon form(back substitution can be used to solve for variables, if necessary), Gauss-Jordan continues until the matrix is in reduced row echelon form.