Is there a formula that gives me the RREF of a matrix?

[Rorschach]

Is there formula that transforms a matrix into its row-reduced echelon form?

I know I can get there by row operations. But isn't there be like a formula?

 

[S.G. Janssens]

One possibility is to write $E_A = E_r E_{r-1} \cdots E_1 A$ where $E_A$ is the RREF of $A$ and $E_1,\ldots,E_r$ are the matrices corresponding to the elementary row operations that transform $A$ into $E_A$. Of course, the particular sequence of row operations depends on $A$, so I don't know if this is what you are looking for, but at least it gives you an equality between matrices.

 

[I like Serena]

Hi Rorschach,

It may seem a bit lame, but a formula to find the RREF of a matrix $A$ would be $\operatorname{rref}(A)$.
The function $\operatorname{rref}$ is supported by for instance the numerical programs MatLab and Octave, while Mathematica has named the function $\operatorname{RowReduce}$.
It can be calculated with the usual Gaussian elimination, but that is not necessarily the best way to do it. The reason is that Gaussian elimination is sensitive to rounding errors. Btw, in practice we generally wouldn't use RREF. Instead we would use a solution method that is the most appropriate for the actual problem that we want to solve.