next up previous contents index
suivant: Construction of a Galois monter: Computing in /p or précédent: Inverse of a matrix   Table des matières   Index

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) :

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 :
are the same as the solutions of:
Example, solve in $ \mathbb {Z}$/13$ \mathbb {Z}$

$\displaystyle \left\{\vphantom{\begin{array}{lcr}\ \ x\ +\ \ 2 \cdot y & = &9 \\  3 \cdot x +10 \cdot y & =& 0 \end{array}}\right.$$\displaystyle \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 :
hence x=3%13 and y=3%13.

giac documentation written by Renée De Graeve and Bernard Parisse