Row reduction to echelon form in ℤ/pℤ: Rref

5.35.7 Row reduction to echelon form in ℤ/pℤ: Rref

In Xcas mode, Rref is simply the inert form of rref; namely, it returns rref without evaluating it. (See Section 5.56.3.) In Maple mode, the Rref command can additionally be used in conjunction with mod to find the reduced row echelon form of a matrix whose elements are in ℤ/pℤ.

(In Maple mode.)
Rref takes one argument:
A, a matrix with elements in ℤ/pℤ.
Rref(A) returns the reduced row echelon form of A.

Example

Solve in ℤ/13ℤ:

⎧
⎨
⎩

x + 2 · y	=	9
3 · x +10 · y	=	0

Input (in Xcas mode):

Rref([[1,2,9] mod 13,[3,10,0] mod 13])

Output:

rref

⎛
⎜
⎜
⎜
⎝

⎡
⎢
⎢
⎢
⎣

1%13

2%13

⎛
⎝

−4

⎞
⎠

%13

3%13

⎛
⎝

−3

⎞
⎠

%13

0%13

⎤
⎥
⎥
⎥
⎦

⎞
⎟
⎟
⎟
⎠

To actually get the reduced echelon form, you can enter:
Input:

eval(ans())

Output:

⎡
⎢
⎣

1%13	0%13	3%13
0%13	1%13	3%13

⎤
⎥
⎦

and conclude that x=3%13 and y=3%13.
Input (in Maple mode):

Rref([[1,2,9],[3,10,0]] mod 13)

Output:

⎡
⎢
⎣

1	0	3
0	1	3

⎤
⎥
⎦

and again conclude that x=3%13 and y=3%13.