Returning the matrix which represents the row reductions in Gauss-Jordan Elimination

I have a square matrix A. I want to row reduce it so it is in Upper Triangular form. When row reducing you perform operations which can also be represented with matrices that are just multiplied to the matrix A. I want to supply the matrix A and have returned the reduced matrix E and the matrix which represents all of the row operations R such that RA=E.

I can do this by hand, and spent too much time on a 9x9 matrix, but I would like to check my work. I just don't know what this is called. I'm not decomposing the matrix and RowReducing just returns the completed matrix.

• No, not an LU Decomposition. That would be A=LU. I want RA=E. Where A is the original matrix, E is the row-reduced matrix, and R is the product of the elementary matrices which perform the reducing row operations. That way when I multiple RA it returns the row-reduced form of matrix A. – MCSquared78 Sep 10 '16 at 2:51

Here is an example of how $R$ and $E$ could be calculated given $A$. This example uses lower case variables and an singular 3x3 matrix $A$, just to see what happens. Try it with your 9x9 matrix.