# Thread: Solve a system using Gauss-Jordan elimination.

1. ## Solve a system using Gauss-Jordan elimination.

So I started this problem but I am now completely stuck. I am supposed to first turn it into reduced row echelon form, however I just don't know where to go from my last step since everything on the bottom is now 0. I am trying to turn the bottom 0 in column 3 to a 1 to make it a leading 1 however I don't know how to go about doing it without messing up my previous work

2. ## Re: Solve a system using Gauss-Jordan elimination.

You can literally just read off your matrix and put your variables back in. It looks like you row reduced correctly.

$\begin{Bmatrix} x_{1} - x_{3} = 2 \\ x_{2} + x_{3} = 7 \end{Bmatrix}$

Now you have two equations that depend on the variable $x_{3}$.

Note that this only works with zeroes filling your bottom row because the last entry (the number that ends up on the right of the equals sign) is also zero. If the last entry was a 5, then the bottom row would read 0 = 5, which is of course bogus.

3. ## Re: Solve a system using Gauss-Jordan elimination.

You say I row reduced correctly but I don't understand how it is in reduced row echelon form? In the 3rd column there is a non zero under the 1, doesn't that mean it isn't reduced?

4. ## Re: Solve a system using Gauss-Jordan elimination.

No, you don't need a pivot in every row for it to be in reduced row echelon form.

The three requirements are (pasted from Wikipedia):
- All nonzero rows (rows with at least one nonzero element) are above any rows of all zeroes
- The leading coefficient (the first nonzero number from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it.
- Every leading coefficient is 1 and is the only nonzero entry in its column

5. ## Re: Solve a system using Gauss-Jordan elimination.

Hmm, what exactly is a "pivot"? I thought if there is a 1 in a column it is automatically a leading 1.

6. ## Re: Solve a system using Gauss-Jordan elimination.

A pivot is an element in the matrix in which all of the entries above and below it are 0. Looking at your final reduced matrix, there are two pivots, one in each of the first two columns.

7. ## Re: Solve a system using Gauss-Jordan elimination.

I see, so a pivot is basically a leading 1? Also, I thought if there was a 1 in a column it is automatically considered a leading 1, and above and below would have to be turned into zeroes :S