*Gaussian elimination, also known as row reduction, is an algorithm in linear algebra for solving a system of linear equations.*

So let me replace the third row with the third row minus 2 times the second row. 3 minus 2 times 2, that's 3 minus 4, or minus 1.

2 minus 2 times 1, that's 2 minus 2, that's 0.

So what's the augmented matrix for this system of equations?

So the first thing, I have a leading 1 here that's a pivot entry.

It's to the right of this one, which is what I want for reduced row echelon form.

And to zero this guy out, what I can do is I can replace the first row with the first row minus the second row.

The second part (sometimes called back substitution) continues to use row operations until the solution is found; in other words, it puts the matrix into reduced row echelon form.

Another point of view, which turns out to be very useful to analyze the algorithm, is that row reduction produces a matrix decomposition of the original matrix.

The process of row reduction makes use of elementary row operations, and can be divided into two parts.

The first part (sometimes called forward elimination) reduces a given system to row echelon form, from which one can tell whether there are no solutions, a unique solution, or infinitely many solutions.

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