I have another question: Let's say we have
R1, R2, R3 (representing scalar equations on a plane) and we perform for example
R2 = R3-R1
So, we now have R1,R2',R3
Can we then do this: R3=R2' - 3R1 ? or we can not use R2' ever again?
You can use R2' after this. What you can't use is R2.
I shouldn't say can't. R2 is still a valid equation and as such you can add multiples of it to the other equations. But if you are doing things correctly you shouldn't want to.
Generally at the kth step you want to use the normalized kth row, which now has 0's in columns 1, k-1, and a 1 at column k, to zero out the k+1, nth rows.
So you normalize row 1, zero out column 1 in rows 2 and 3.
Then normalize row 2. Zero out the 2nd column in row 3.
Normalize row 3 and you are in upper triangular form and can back substitute.
This works fine however what you might also do, to avoid the fractions getting too obnoxious from normalization, is swap rows. Especially if a row already has a 1 in the column you are about to zero out. In that case you don't need to normalize it.
Do you have to normalize during each step? No. You can leave normalization to the end and add or subtract multiples of your "pivot row" (the row that's being used to zero out the others) to a multiple of the row being zeroed and there are actually good reasons numerically to do this. But if your row is all multiples of the pivot element then by all means normalize it.
Your best bet to learn this is to write up a small program to toss 3x3 or 4x4 matrices at you to row reduce and solve. Then check the answers but plugging the solutions back in. Keep at this until you start to get every one correct.
This diatribe probably leaves you more confused than you were. Maybe look at this.
Thank you for that resource it is very helpful!
Edit: As part of my Independent study, I do not see how Matrixes are any useful or better than solving the intersection of planes or lines in 3D or 2D. So why bother with Matrix? Is there any long term benefit?
Since I couldn't edit my previous post:
If my 3rd row in my matrix is for example [0 0 18 | 36] do I have to draw another Matrix and show the simplification [0 0 1 | 2] or can I just algebraically? Is it considered a communication error If I do not show the simplification in a matrix format?
and Is my format correct here, Also can I simply the equations before starting solving my augmented matrix like I did here: