rref find the row reduction to echelon form of a matrix with coefficients in ℤ/pℤ.
This may be used to solve a linear system of equations with coefficients in ℤ/pℤ, by rewriting it in matrix form (see also 2.52.3) :
rref takes as argument the augmented matrix
of the system (the matrix obtained by augmenting matrix A to the
right with the column vector B).
rref returns a matrix [A1,B1] : A1 has 1 on
it’s principal diagonal, and zeros outside, and the
solutions in ℤ/pℤ, of :
are the same as the solutions of:
Example, solve in ℤ/13ℤ
⎧ ⎨ ⎩ 

Input :
Or :
Output :
hence x=3%13 and y=3%13.