Can someone explain the gauss-jordan method in baby steps?

Question

I know how to set up a matrice ...that's it though. Please show me every single little step...even if it seems unimportant. Use like a 5th grade vocabulary too please. No fancy math terms. I'm trying to understand these before my teacher starts showing them to us, because I am....is it called innumerate? Ok, w/e, I'm stupid. Thanx!

Helmut, thank you soooo much. No one wanted to teach me matrices, because of the time it takes to explain them. I wasn't sure if anyone would bother to take the time here to explain them either. Your answer is worth gold to me. I don't have gold so I'm giving you the 10 points. Thanx for being so nice and taking the time!

Answer 1

To explain it without using a whole lot of confusing letters, we need to have 3 equations in 3 unknowns that are solvable. I cheated & made up 3 that have small cooefficients and integer solutions:
3x + 5y + 2z = 22
1x + 2y - 1z = 8
-2x + 5y + 3z = 2
The goal is to arrive at the matrix
1 0 0 x
0 1 0 y
0 0 1 z
First, set up the matrix:
3 5 2 22
1 2 1 8
-2 5 3 2
The easiest way to get a 1 in the top left row is exchanging rows:
1 2 1 8
-2 5 3 2
3 5 2 22
Now we want 0's in the 1st position of the 2nd & 3rd row (eliminating x), so we multiply the first row by the negative of the 1st number in row 2, and add to get
2 4 2 16
-2 5 3 2
------------
0 9 5 18
giving us
1 2 1 8
0 9 5 18
3 5 2 22
Doing the same with rows 1 & 3:
-3 -6 -3 -24
3. 5. 2.. 22
------------
0 -1 -1 -2
giving
1 2 1 8
0 9 5 18
0 -1 -1 -2
If we multiply row 3 by -1 and swap rows 2&3, we get
1 2 1 8
0 1 1 2
0 9 5 18
Now we need to combine rows 2 &3 to get the form 0 0 n n
0-9-9-18
0 9 5 18
------------
0 0 -4 0
dividing by -4,
0 0 1 0
giving
1 2 1 8
0 1 1 2
0 0 1 0 (z = 0)
From this point we "back substitute" (substitute z into row 2 & solve for y)
0 1y + 1*0 = 2
0 1y + 0 = 2
giving
1 2 1 8
0 1 0 2 (y = 2)
0 0 1 0 (z = 0)
Finally, back-substitutng into row 1 (solving for x):
1x + 2*2 + 1*0 = 8
1x + 4 + 0 = 8
1x 0 0 = 4
giving the final form
1 0 0 4 (x = 4)
0 1 0 2 (y = 2)
0 0 1 0 (z = 0)

I see now that I shouldn't have manufactured equations with a 0 solution for one variable. It makes the process look more contrived than it is.

Answer 2

Hehehe. Nobody is born knowing this stuff (Not even me ☺)

Gauss-Jordan is simply another way to solve a system of linear equations. It's a bit long and detailed to show it here, but there is a fairly good discussion of it at
http://www.convict.lu/Jeunes/Math/Gauss_Jordan_Elimination.htm
and there are a lot of other sites on the web that also discuss it.


Doug

Answer 3

Hey. There is a little mistake in Helmuts matrix, in the second file it should have been -1, not 1

Ana