Row reduction to echelon form in $\mathbb {Z}$/p$\mathbb {Z}$ : rref

rref find the row reduction to echelon form of a matrix with coefficients in $\mathbb {Z}$/p$\mathbb {Z}$.

This may be used to solve a linear system of equations with coefficients in $\mathbb {Z}$/p$\mathbb {Z}$, by rewriting it in matrix form (see also 1.52.3) :

A*X=B

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 $\mathbb {Z}$/p$\mathbb {Z}$, of :

A1*X=B1

are the same as the solutions of:

A*X=B

Example, solve in $\mathbb {Z}$/13$\mathbb {Z}$

$$\left\{\vphantom{\begin{array}{lcr}\ \ x\ +\ \ 2 \cdot y & = &9 \\ 3 \cdot x +10 \cdot y & =& 0 \end{array}}\right.$$$$\begin{array}{lcr}\ \ x\ +\ \ 2 \cdot y & = &9 \\ 3 \cdot x +10 \cdot y & =& 0 \end{array}$$

Input :

rref([[1, 2, 9]%13,[3,10,0]%13])

Or :

rref([[1, 2, 9],[3,10,0]])%13

Output :

[[1%13,0%13,3%13],[0%13,1%13,3%13]]

hence x=3%13 and y=3%13.