I have the augmented matrix
$$\begin{bmatrix} 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1\\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ \end{bmatrix}$$
And I need to reduce it to reduced echelon form over a field of three elements ($\mathbb{F}_{3}$)
If I reduce the matrix over $\mathbb{Q}$ I get:
$$\begin{bmatrix} 1 & 0 & 0 & 2\\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 3 \\ \end{bmatrix}$$
I do get the concept of field, but I have no idea how apply it in order to reduce the matrix over a field of 3 elements. Do I have to make it e.g. 3=3-3=0 (Subtract by the number of elements in the field)?
Thank you in advance for your help!