Enter Matrix and Pivot Settings
Use decimals or fractions such as 1/2. Separate values with commas, spaces, semicolons, or vertical bars. Row and column numbers are 1-based.
Example Data Table
| # | C1 | C2 | C3 | C4 |
|---|---|---|---|---|
| R1 | 2 | 1 | -1 | 8 |
| R2 | -3 | -1 | 2 | -11 |
| R3 | -2 | 1 | 2 | -3 |
Chosen pivot: row 1, column 1.
| # | C1 | C2 | C3 | C4 |
|---|---|---|---|---|
| R1 | 1 | 0.5 | -0.5 | 4 |
| R2 | 0 | 0.5 | 0.5 | 1 |
| R3 | 0 | 2 | 1 | 5 |
Effect: the pivot becomes 1, and the other values in that pivot column become 0.
Formula Used
1) Normalize the pivot row
Rpnew = Rp / ap,c
2) Eliminate the remaining entries in the pivot column
Rinew = Ri - ai,c × Rpnew, for all i ≠ p
Here, ap,c is the selected pivot element. After the operation, that pivot entry becomes 1, and all other entries in the same column become 0.
How to Use This Calculator
- Paste or type your matrix with one row per line.
- Separate values with commas, spaces, semicolons, or vertical bars.
- Enter the pivot row and pivot column using 1-based indexing.
- Choose how many decimals you want in the displayed result.
- Click Calculate Pivot Operation to generate the transformed matrix.
- Review the original matrix, pivoted matrix, and row-operation steps.
- Use the CSV or PDF buttons to export the current result.
- Check the heatmaps to compare how the matrix changed after pivoting.
FAQs
1) What is a pivot operation in matrix algebra?
A pivot operation selects a non-zero entry, scales its row so the entry becomes 1, then eliminates the other values in that column. It is a core step in Gauss-Jordan elimination, simplex methods, and matrix inversion workflows.
2) Why must the chosen pivot be non-zero?
Division by zero is undefined, so a zero pivot cannot normalize its row. You must either choose another pivot location or swap rows first, then perform the pivot step on a non-zero entry.
3) Can I use fractions instead of decimals?
Yes. The calculator accepts fractions such as 1/2, -3/4, or 5/8. It converts them to decimal values internally, performs the pivot operation, and then displays the result using your selected decimal precision.
4) Does this work for augmented matrices and tableaux?
Yes. You can enter ordinary matrices, augmented systems, or simplex-style tableaux. The tool treats each row consistently and applies the same pivot logic across all columns in the selected structure.
5) Why do some outputs show tiny decimal residues?
Floating-point arithmetic can create very small rounding artifacts, especially after repeated subtraction. This calculator suppresses tiny values near zero, but slight decimal remnants may appear if your matrix contains repeating or non-terminating values.
6) What does the Plotly graph represent?
The heatmaps show the original matrix and the pivoted matrix as color-coded value grids. They help you compare magnitude changes, identify the normalized pivot row, and visually confirm elimination throughout the pivot column.
7) When is a pivot operation especially useful?
It is useful in solving linear systems, reducing matrices to row-echelon forms, computing inverses, and running simplex iterations in optimization. Any workflow that relies on systematic row operations can benefit from pivot steps.
8) Does the calculator swap rows automatically?
No. This page performs the pivot step only on the pivot position you select. If your chosen pivot entry is zero, change the pivot location or manually reorder rows in the input before calculating.