# Row reduction of a matrix over $\mathbb Z_{3}$ (verification)

Row reduce the following matrix over $\mathbb Z_{3}$ to row-reduced echelon form:

$$M= \left[ {\begin{array}{cc} 1 & 2 & 1 \\ 1 & 1 & 1 \\ 2 & 2 & 1 \\ \end{array} } \right]$$

\begin{array}{cc} 1 & 2 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \\ \end{array}

but when using Mathematica to verify my answer, it gave me this answer:

\begin{array}{cc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}

Is my answer wrong, or are these equivalent?

UPDATE

So if I do R1 $\rightarrow$ R1 - R3 I get:

\begin{array}{cc} 1 & 2 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \\ \end{array}

Is this still not in reduced row echelon form? If not, why?

• I updated my answer, is this still not in row reduced echelon? – user1282637 Sep 8 '14 at 1:46

• So if I do R1 $\rightarrow$ R1 - R3 I get: \begin{array}{cc} 1 & 2 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \\ \end{array} Is this still not in reduced row echelon form? – user1282637 Sep 8 '14 at 1:44