Calculator Input
Use sparse triplets or dense rows. The form uses responsive columns for large, medium, and mobile screens.
Plotly Graphs
The first chart shows matrix values as a heatmap. The second chart compares nonzero counts across rows and columns.
Example Data Table
This example matches the prefilled data. It creates a sparse 4 × 4 matrix with determinant 242.
| Row | Column | Value | Comment |
|---|---|---|---|
| 1 | 1 | 5 | Leading diagonal entry |
| 1 | 2 | 1 | Upper off-diagonal value |
| 2 | 1 | -2 | Lower off-diagonal value |
| 2 | 2 | 4 | Second diagonal entry |
| 3 | 3 | 3 | Third diagonal entry |
| 3 | 4 | 7 | Upper block value |
| 4 | 3 | 1 | Lower block value |
| 4 | 4 | 6 | Final diagonal entry |
Formula Used
The calculator transforms the matrix into an upper triangular matrix using Gaussian elimination. For a square matrix A, the determinant is:
det(A) = s × Π Uii
Here, s is the sign change caused by row swaps, and Uii are the diagonal entries of the upper triangular matrix U. If any pivot becomes zero within the chosen tolerance, the matrix is treated as singular and the determinant is zero.
Sparse matrices are stored efficiently because only nonzero values matter during input. The determinant itself still depends on the full square structure, so elimination is performed on the assembled matrix after parsing sparse entries.
How to Use This Calculator
- Choose the matrix size from 2 to 10.
- Select sparse triplets or dense rows.
- Enter values using 1-based row and column indexing for triplets.
- Adjust tolerance and display precision if needed.
- Choose partial pivoting for better numerical stability.
- Press Calculate Determinant to view the result above the form.
- Review the original matrix, triangular matrix, pivots, density, and charts.
- Use the CSV or PDF buttons to export the report.
FAQs
1. What is a sparse matrix?
A sparse matrix contains many zero entries and relatively few nonzero values. Sparse storage is useful because it reduces manual input and highlights structure clearly.
2. Why use triplet input for sparse matrices?
Triplet input lets you enter only row, column, and value for nonzero positions. That is faster than typing every zero in a large matrix.
3. When does the determinant become zero?
The determinant becomes zero when the matrix is singular. In practice, that means elimination finds a pivot that is effectively zero under the selected tolerance.
4. Why is pivoting important?
Pivoting improves numerical stability by choosing a stronger pivot in each column. That reduces rounding problems and makes determinant estimates more reliable.
5. Does a sparse matrix always have a small determinant?
No. Sparsity describes how many zeros appear, not the magnitude of the determinant. A sparse matrix can have a large, small, or zero determinant.
6. Can I paste a full dense matrix instead?
Yes. Switch the input mode to dense rows and paste one row per line. The calculator will parse the full square matrix directly.
7. What does density mean here?
Density is the percentage of nonzero entries compared with all positions in the matrix. Lower density means the matrix is more sparse.
8. What do the exports include?
The exports include determinant summary values, sparsity statistics, pivot data, and the original matrix. The PDF is formatted for reporting and printing.