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In any event, you can read more about Gaussian elimination here:

http://en.wikipedia.org/wiki/Gaussian_elimination

Basically, though, you can take a row in an augmented matrix and do a few different things to it.

1) Multiply it by a constant.

2) Switch it with another row.

3) Add another row to it.

Gaussian elimination just consists in performing a set of these "row operations" such that your matrix is reduced to "echelon form", where all nonzero rows are above all zero rows, and the first nonzero number in each row is a) 1, b) in a column further to the right than the row above it.

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HallsofIvy

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[tex]\begin{bmatrix}a & b & c \\ d & e & f\end{bmatrix}[/tex]

to

[tex]\begin{bmatrix}1 & 0 & p \\ 0 & 1 & q\end{bmatrix}[/tex]

That is, you want the first number in the first column (more generally the numbers on the main diagonal) to be

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For the equations ax+by = c, and px + qy = d

a b : c

p q : d

Creating some sort of row operation, for example: If a=3 and p=3, then the operation would be Row 1 - Row 2. Then, the new augmented matrix is

0 b-q : c-d

p q : d

And, the new equation to determine the value of Y is (b-q)y = (c-d) and solving for y is simple at that point. To determine the value of x, the book then goes on to demonstrate that one only need plug in the value for y, and solve for x in the px + qy = d equation.

I'm aware that there is a method for using the identity matrix, but my book isolates that completely from the method for Gaussian elimination.

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vela

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Gauss-Jordan elimination goes a bit further and transforms the coefficient matrix into the identity matrix.

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