### 2.31.17 Row reduction to echelon form in ℤ/pℤ : `rref`

`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) :

`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 ℤ/pℤ, of :

`A1*X=B1`

are the same as the solutions of:

`A*X=B`

Example, solve in ℤ/13ℤ

⎧ ⎨ ⎩ |
x + 2 · y | = | 9
3 · x +10 · y | = | 0

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`.