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Row Reduction

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Elementary Row Operations

We can perform three elementary row operations on matrices:

- Multiplying a row by a constant.
- Switching two rows.
- Adding a constant times a row to another row.

*Examples*:

- Multiplying a row by a constant:

- Switching two rows:

- Adding a constant times a row to another row:

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Row Reduction

We perform row operations to row reduce a matrix; that is, to
convert the matrix into a matrix where the first *m*×*m* entries
form the identity matrix:

where

* represents any number.

This form is called reduced row-echelon form.

*Note:* Reduced row-echelon form does not always produce the
identity matrix, as you will learn in higher algebra. For our purposes,
however, we will consider reduced row-echelon form as only the form in
which the first *m*×*m* entries form the
identity matrix.

To row reduce a matrix:

- Perform elementary row operations to yield a "1" in the first row,
first column.
- Create zeros in all the rows of the first column except the first
row by adding the first row times a constant to each other row.
- Perform elementary row operations to yield a "1" in the second row,
second column.
- Create zeros in all the rows of the second column except the second
row by adding the second row times a constant to each other row.
- Perform elementary row operations to yield a "1" in the third row,
third column.
- Create zeros in all the rows of the third column except the third
row by adding the third row times a constant to each other row.
- Continue this process until the first
*m*×*m* entries form the
identity matrix.