### 11.9.7 Row reduction to echelon form in ℤ/pℤ

In Xcas mode, Rref is simply the inert form of rref; namely, it returns rref without evaluating it (see Section 15.7.3). 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 A, a matrix with elements in ℤ/pℤ.
- Rref(A) returns the reduced row echelon form of A.

#### Example

Solve in ℤ/13ℤ:

⎧
⎨
⎩

x | + | 2y | = | 9,
3x | + | 10y | = | 0.

Input in Xcas mode:

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

rref

⎛
⎜
⎜
⎜
⎝

⎡
⎢
⎢
⎢
⎣

1%13

2%13

⎛ ⎝ | −4 | ⎞ ⎠ | %13

3%13

⎛ ⎝ | −3 | ⎞ ⎠ | %13

0%13

⎤
⎥
⎥
⎥
⎦

⎞
⎟
⎟
⎟
⎠

To actually get the reduced echelon form, enter:

eval(ans())

⎡
⎢
⎣

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

⎤
⎥
⎦

Input in Maple mode:

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

⎡
⎢
⎣

1 | 0 | 3
0 | 1 | 3

⎤
⎥
⎦

In both cases you conclude that x=3%13 and y=3%13.