Linear Algebra, simplifying large matrices

[maximade]

Homework Statement

Find all solutions using Gauss-Jordan elimination:
[ 0 0 0 1 2 -1 l 2
1 2 0 0 1 -1 l 0
1 2 2 0 -1 1 l 2]

Homework Equations

Switching rows,
able to scale any row
able to add non zero multiple to row

The Attempt at a Solution

What I did was switch the 1st and 3rd row, then i tried my best to get it as close to rref form.
I ended up getting:
[ 1 2 0 0 1 -1 l 0
0 0 -2 0 2 -2 l -2
0 0 0 1 2 -1 l 2]

I honestly can't think of a way to reduce it anymore, not to mention that I have no idea how to even put this in "solution" form.

 

[Mark44]

Homework Statement

Find all solutions using Gauss-Jordan elimination:
[ 0 0 0 1 2 -1 l 2
1 2 0 0 1 -1 l 0
1 2 2 0 -1 1 l 2]

Homework Equations

Switching rows,
able to scale any row
able to add non zero multiple to row

The Attempt at a Solution

What I did was switch the 1st and 3rd row, then i tried my best to get it as close to rref form.
I ended up getting:
[ 1 2 0 0 1 -1 l 0
0 0 -2 0 2 -2 l -2
0 0 0 1 2 -1 l 2]

I honestly can't think of a way to reduce it anymore, not to mention that I have no idea how to even put this in "solution" form.

So far, so good, but you can go one step further by replacing the 2nd row by (-1/2) of itself.

That gives you
[1 2 0 0 1 -1 l 0]
[0 0 1 0 -1 1 l 1]
[0 0 0 1 2 -1 l 2]

Now it's in RREF form, with each leading entry being 1.

What you have is a matrix that represents three equations in six variables. A system like this typically has an infinite number of solutions.

To write the solution, solve for each variable that has a leading entry: x₁, x₃, and x₄. The other three variables are free variables that can have any value.

 

[maximade]

Cool, I didn't know that the rref form doesn't need to have straight diagonal "1"s.

As for the solution, would it simply be:
a+2b+e-f=0
c-e+f=1
d+2e-f=2?

 

[Mark44]

I would go further and solve for the first variable in each of your equations.

a = -2b - e + f
c = e - f + 1
d = -2e + f + 2
b, e, f are arbitrary