Solving Systems: Row Operations

1. Elementary Row Operations

To solve a system using a matrix, we need to simplify it. We do this by replacing the original system with a new, easier equivalent system. There are exactly three legal moves we can make, called elementary row operations:

• Multiply: Multiply a whole row by a non-zero constant.
• Swap: Interchange any two rows.
• Add: Add a constant times one row to another row.

These operations change the numbers, but they do not change the solution to the system!

2. Row Echelon Form (REF) and Reduced Row Echelon Form (RREF)

Our goal with elementary row operations is to simplify the matrix until the solution is obvious. We do this by getting the matrix into a specific staggered shape.

Row Echelon Form (REF)

REF requires three rules:

1. If a row is not entirely zeros, its first non-zero number from the left must be a 1 (called a "leading 1").
2. Any rows that are entirely zeros must be grouped together at the very bottom of the matrix.
3. As you read down the rows, the leading 1 of a lower row must be strictly further to the right than the leading 1 of the row above it (creating a staircase pattern).

Reduced Row Echelon Form (RREF)

RREF is the "ultimate" simplified state. It must follow the three rules of REF, plus a strict fourth rule:

4. Each column that contains a leading 1 must have zeros everywhere else in that entire column (both above and below the leading 1).

3. The Gauss-Jordan Method

The Gauss-Jordan Method is the formal process of taking an augmented matrix and using elementary row operations to completely reduce it into Reduced Row Echelon Form (RREF). Once in RREF, you simply rewrite the system into equations to find your solution.

Example

If your RREF matrix looks like this:

$$\left[\begin{array}{ccc|c} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2 \end{array}\right]$$

It simply translates to $x = 3$, $y = 1$, and $z = 2$.