Title: Row Reduction Method

The principles involved in row reduction of matrices are equivalent to those we used in the elimination method of solving systems of equations. That is, we are allowed to

1.Multiply a row by a non-zero constant.

2.Add one row to another.

3.Interchange between rows

4.Add a multiple of one row to another.

How do we use this system to solve systems of equations? We follow the steps:

1. Write the augmented matrix ofthe system.

2.Row reduce the augmented matrix.

3.Write the new, equivalent, system that is defined by the new, row reduced, matrix.

4.The solution (or lack thereof) is now apparent!

Example:

Let us use row reduction to solve the system of equations

3x + 2y - z = 1

x - 2y + z = 0

2x + y - 3z = -1

Step 1:

Write the augmented matrix of the system:

Step 2:

Row reduce the augmented matrix:

The symbols we used above the arrows are short for:

R1 <---> R2

Interchange Rows 1 and 2.

R2 - 3R1

Subtract 3 times (new) Row 1 from Row 2.

R2 / -4

Divide Row 2 by -4.

3R3 / 2

Multiply row 3 by 3/2.

Step 3:

Rewrite the system using the row reduced matrix:

x + 2y + z = 0

y + z = -1/4

z = 7/8

Step 4:

And the solution is found by going from the bottom equation up:

z = 7/8

y = (-1/4) - (7/8) = - 9/8

x = 0 - (7/8) - 2(-(9/8)) = 11/8

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