## Row reduction

### January 4, 2008

### What is row reduction?

* Row reduction* (or * Gauss-Jordan elimination*) is the process of using * row operations* to reduce a matrix to * row reduced echelon form*. This procedure is used to solve systems of linear equations, invert matrices, compute determinants, and do many other things.

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A matrix is in * row reduced echelon form* if the following conditions are satisfied:

- The first nonzero element in each row (if any) is a “1” (a
*leading coefficient*). - Each leading coefficient is the only nonzero element in its column.
- All the all-zero rows (if any) are at the bottom of the matrix.
- The leading coefficients form a “stairstep pattern” from northwest to southeast:

In this matrix, the leading coefficients are in positions , , , … .

Here some more matrices in row-reduced echelon form. Notice the “stairstep pattern” made by the leading coefficients. The *’s indicate that the numbers in those positions can be anything.

### Example

is in row reduced echelon form. This matrix is called the * identity matrix*.

…

is not in row reduced echelon form. The first nonzero element in row 2 is a “7”, rather than a “1”.

…

is in not row reduced echelon form. The leading coefficient in row 3 is not the only nonzero element in its column.

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is not in row reduced echelon form. There is an all-zero row lying above a nonzero row.

…

is not in row reduced echelon form. The leading coefficient in row 2 is not the only nonzero element in its column.

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Useful link for *Row-reducting Algorithm*:

http://marauder.millersville.edu/~bikenaga/linearalgebra/rowred/rowred.html