Linear Algebra and Matrix Theory

Question

Given the matrix $A$ listed below as a matrix over field $z\in \{0,1,2,3,4\}$, find the row reduced echelon form $B$ of $A$. List the elementary matrices used to reduce $A$ to $B$. $$A=\pmatrix{1 &2 &0 &3 \\ 2 &4 &1 &1 \\ 2 & 4 &0 &1 \\}$$

I am able to get the Matrix into the reduced row echelon form, the problem is that when I am getting my elementary matrices the way I reduce Matrix $A$ always makes it so my elementary matrices are not in the field. Please help me.

One way I tried was $$R_2 \leftarrow R_2-R_3$$ and $$R_3 \leftarrow 2R_1-R_3$$ that gets the matrix into reduced row echelon from but puts the elementary matrices outside of the field.

Answer 1

$$\begin{pmatrix}1&2&0&3\\2&4&1&1\\2&4&0&1\end{pmatrix}\stackrel{R_2+3R_1}{\stackrel{R_3+3R_1}\longrightarrow}\begin{pmatrix}1&2&0&3\\0&0&1&0\\0&0&0&0\end{pmatrix}$$

and we're done working over $\,\Bbb F_5:=\Bbb Z/5\Bbb Z\,$ . Can you now list the elementary matrices used, even if the first, and only, used operation is subdivided in two?