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

Im not sure how to go about answering this question please help point me in the right direction.

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We really don't know what you know or don't know, or what you have or have not tried. This means that we can't really help you without giving you the answer outright. What have you tried so far? – mixedmath Jan 11 '13 at 5:01
well I was thinking that the number of different matrices depended solely on the module being used. I thought the answer might be 1.{0,1,2} 2. {1,2,0} 3. {2,0,1), where they are supposed to under each other to form a matrix. so only one but i dont think thats right. Im just a little confused as to what i need to show or prove – user547866 Jan 11 '13 at 5:08

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 $2$.

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i thought for reduced row echelon form only the pivot numbers could be nonzero and everything above and below had to be 0. – user547866 Jan 11 '13 at 5:15
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

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