Transform the following $3 \times 3$ matrix $A$ with entries in $\mathbb{Z}/5\mathbb{Z}$ into reduced row echelon form (with $[1]$ as Pivot element and $[0]$ above the Pivot element). State out the Modulo elements through representatives $x$ with $0 \leq x \leq 4$.
$A =: \begin {pmatrix} [1] & [1] & [2]\\ [0] & [2] & [4] \\ [2] & [4]& [3] \end{pmatrix}$
My idea:
$\begin {pmatrix} 1 & 1 & 2\\ 0 & 2 & 4 \\ 2 & 4& 3 \end{pmatrix} (III-2I) = \begin {pmatrix} 1 & 1 & 2\\ 0 & 2 & 4 \\ 0 & 2 & 4 \end{pmatrix} III-II = \begin {pmatrix} 1 & 1 & 2\\ 0 & 2 & 4 \\ 0 & 0 & 0 \end{pmatrix} II \cdot [3] = \begin {pmatrix} 1 & 1 & 2\\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{pmatrix}~~~~~~~~ I-II = \begin {pmatrix} 1 & 0 & 0\\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{pmatrix}$
How to go on?