Row Reduction over indicated field

[ur5pointos2sl]

Hi-

I am taking linear algebra and the teacher threw some problems at us that I cannot seem to figure out or find examples for online. Also, the textbook doesn't even cover the material.

The problem states:
Using Gauss-Jordan elimination, solve the following system with coefficients in indicated field.

In Z₃:

2x + y = 1
x + y = 2

Now I understand completely how to row reduce this system, but am not sure how to proceed in Z₃.

Any help appreciated. Thanks!

 

[micromass]

The best thing to do here is to solve the system as you normally would and then interpret the solution in $\mathbb{Z}_3$.

So, how would you proceed normally?? Just write the first step, don't solve it completely yet.

 

[ur5pointos2sl]

Thanks for the reply.

The first thing I would do is swap row 1 and row 2 to get a leading 1 in row 1.

[1 1 l 2]
[2 1 l 1]

From here would take R2-2R1.

 

[micromass]

OK, so do R2-2R1. But be aware that the arithmetic is in $\mathbb{Z}_3$.

So, for example, the second column gives us 1-2*1=-1=2.

 

[Deveno]

or, (to keep it straight as you go) realize that -2 in Z₃ is actually 1, so instead of R2-2R1, use R2+R1 (which gives you 3 = 0 in the first column).

that is, instead of aiming to "zero out" entries, you just want to get them to the nearest multiple of 3 (which amounts to the same thing in Z₃). with a little practice, you can see how to do this and always keep all the entries positive, which might help a bit.

 

[ur5pointos2sl]

OK, so do R2-2R1. But be aware that the arithmetic is in $\mathbb{Z}_3$.

So, for example, the second column gives us 1-2*1=-1=2.

or, (to keep it straight as you go) realize that -2 in Z₃ is actually 1, so instead of R2-2R1, use R2+R1 (which gives you 3 = 0 in the first column).

that is, instead of aiming to "zero out" entries, you just want to get them to the nearest multiple of 3 (which amounts to the same thing in Z₃). with a little practice, you can see how to do this and always keep all the entries positive, which might help a bit.

Thanks for the reply. The part I don't understand is how -1 = 2 and -2 = 1 in Z₃. Would you be able to explain this part to me?