RREF and Matrix Rank Calculator
Reduce a 3×3 matrix to reduced row-echelon form and find its rank using Gaussian elimination steps.
Calculate
Enter a 3×3 matrix. Use decimals or integers.
What this calculator teaches
Reduced row-echelon form turns a matrix into a simpler equivalent matrix using row operations. It helps solve systems, identify pivots, and find rank.
Rank counts the number of pivot rows. It measures how many independent rows or columns the matrix has.
Formula & Symbols
| Concept | Formula | Meaning |
|---|---|---|
| Row operation | R_i ← R_i + kR_j | Allowed operation that preserves row equivalence. |
| RREF pivot | leading 1 | Each pivot column has a leading 1 and zeros above and below. |
| Rank | number of pivots | Counts independent rows/columns. |
Step-by-Step Examples
- Rows [1,2,3] and [2,4,6] are multiples.
- Only one of them adds independent information.
- RREF reveals fewer pivot rows.
Where students use this
Common Mistakes to Avoid
Use only valid row operations: swap rows, multiply a row by a nonzero constant, or add a multiple of one row to another.
RREF also requires zeros above each pivot, not just below.
Rank counts pivots, not all nonzero numbers.
Frequently Asked Questions
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Formula Explorer connections
Interpretation: This relationship represents vectors, linear systems or transformations using matrix structure and vector operations. Assumption: Matrix dimensions must be compatible. Inversion, decomposition and unique solutions require rank, conditioning or nonzero-determinant conditions.