Find the Reduced Row Echelon Form (RREF) of a matrix with step-by-step solutions and visualizations.
| 1 | 0 | -1 |
| 0 | 1 | 2 |
| 0 | 0 | 0 |
| 1 | 2 | 3 |
| 0 | -3 | -6 |
| 7 | 8 | 9 |
| 1 | 2 | 3 |
| 0 | -3 | -6 |
| 0 | -6 | -12 |
The Reduced Row Echelon Form (RREF) is a special form of a matrix that is useful for solving systems of linear equations, determining the rank of a matrix, and finding the inverse of invertible matrices.
1. Leading 1s: The first nonzero number in any row is a 1 (called a leading 1 or pivot)
2. Stacked Pivots: Each leading 1 is to the right of the leading 1 in the row above it
3. Zero Rows: Any rows consisting of all zeros are at the bottom of the matrix
4. Pivot Columns: In columns containing a leading 1, all other entries are zero
How to achieve RREF using Gauss-Jordan elimination:
RREF is unique for any given matrix, which means that regardless of the sequence of row operations used, the final RREF will always be the same for a particular matrix.
| Application | How RREF Helps |
|---|---|
| Solving Linear Systems | Directly reveals solutions |
| Matrix Inversion | Finds inverse through augmentation |
| Determining Rank | Count of pivots equals rank |
| Basis Finding | Identifies linearly independent columns |
| Kernel/Null Space | Helps find solution to Ax=0 |
• Engineering: Solving circuit equations
• Economics: Input-output models
• Computer Graphics: Transformation matrices
• Data Science: Dimension reduction
• Physics: Solving equilibrium equations