# Linear algebra and matrix theory

list the different 3x2 row reduced echelon form matrices if the field is F={0,1,2} mod 3.

Hint: Once you decide where to put the leading $1$'s, any other entries that don't have to be $0$ could be $0$, $1$ or $2$.
No, an entry to the right of a pivot entry and in a column that contains no pivot entry can be anything. For example, the $x$'s in $\pmatrix{ 1 & x & 0 & x \cr 0 & 0 & 1 & x\cr}$. –  Robert Israel Jan 11 '13 at 19:37