Overview
This lecture introduces Gauss-Jordan row operations for solving systems of linear equations using augmented matrices, covering the three main types of elementary row operations with examples.
Introduction to Gauss-Jordan Row Operations
- Gauss-Jordan row operations are step-by-step methods used to solve systems of linear equations by simplifying augmented matrices.
- Each row operation produces an equivalent matrix that represents the same system as the previous one.
The Three Elementary Row Operations
- Row Interchange: Swap two rows; notation uses a double arrow indicating two-way exchange (e.g., R1 ↔ R2).
- Row Multiplication: Multiply a row by a non-zero constant and replace it; notation uses a right arrow (e.g., kR3 → R3).
- Row Addition: Add a multiple of one row to another row and put the result in the second row; notation uses a right arrow (e.g., kR1 + R2 → R2).
- Row operations are always performed on rows (not columns), as rows represent equations.
Example Application of Row Operations
- Given an augmented matrix, multiply every entry in row one by one-third and place the result in row one; other rows remain unchanged.
- After the first operation, compute and replace the values in row one with the results.
- For the next step, take twice row one, add it to row two, and put the result in row two, keeping other rows unchanged.
- Perform the arithmetic for each affected row to update the matrix, maintaining system equivalence at each step.
Key Terms & Definitions
- Augmented Matrix — A matrix representing a system of linear equations, combining coefficients and constants.
- Elementary Row Operation — One of three allowed manipulations (row interchange, row multiplication, row addition) to solve systems.
- Equivalent Matrices — Matrices that represent the same system after row operations.
Action Items / Next Steps
- Review and practice Gauss-Jordan row operations on sample augmented matrices.
- Complete any assigned problems on pages 48 and 49 of the lecture notes.