Augmented Matrices

A matrix is simply an organized array of numbers arranged in rows (horizontal) and columns (vertical). We describe a matrix by its dimensions: the number of rows "by" the number of columns. For example, a matrix with 2 rows and 3 columns is a 2×3 matrix.

When solving systems of equations, we use augmented matrices by adding a vertical line to separate the coefficients from the constants. For example, the system of equations x + 3y = 9 and -x + 4y = -2 becomes:

[1  3  |  9]
[-1 4  | -2]

Our goal is to transform augmented matrices into reduced row-echelon form, where:

• Rows with all zeros are at the bottom
• The first non-zero entry in each row is 1
• Each of these leading 1's appears in a different column

💡 Think of row-echelon form as the "solved" version of your system, where each variable is isolated!

To achieve this form, we use Gaussian elimination $also called Gauss-Jordan elimination$, which allows three types of operations:

1. Interchange any two rows
2. Multiply or divide a row by a non-zero constant
3. Add or subtract rows from each other

When a matrix reaches the form where x = -2, y = 1, z = 3, you've successfully solved your system of equations!