Calculator Inputs
Overall page layout stays single-column. Only the calculator fields use the requested responsive grid.
Formula Used
For any permutation matrix P, the determinant equals the sign of the underlying permutation. Two equivalent formulas are used:
1. det(P) = (-1)inv(π)
2. det(P) = (-1)n - c
Here, inv(π) is the number of inversion pairs, n is the permutation size, and c is the total number of cycles including fixed points. Even parity gives determinant 1. Odd parity gives determinant -1.
How to Use This Calculator
- Enter the matrix size n.
- Select either permutation mapping input or direct matrix input.
- Type the permutation as values from 1 to n, or paste the binary matrix.
- Choose whether to display inversion pairs or include fixed points in cycle notation.
- Optionally add an expected determinant for checking.
- Click Calculate Determinant to show results above the form.
- Use the CSV or PDF buttons to export the computed results.
Example Data Table
| Case | Permutation | Cycle Form | Inversions | Parity | Determinant |
|---|---|---|---|---|---|
| Identity | [1, 2, 3, 4] | (1)(2)(3)(4) | 0 | Even | 1 |
| Single swap | [2, 1, 3, 4] | (1 2)(3)(4) | 1 | Odd | -1 |
| 3-cycle | [2, 3, 1, 4] | (1 2 3)(4) | 2 | Even | 1 |
| Reverse 4 | [4, 3, 2, 1] | (1 4)(2 3) | 6 | Even | 1 |
Frequently Asked Questions
1) Why is the determinant only 1 or -1?
A permutation matrix only reorders basis vectors. That preserves lengths and orthogonality, but it may flip orientation. So its determinant must be either 1 for even parity or -1 for odd parity.
2) Can I enter the matrix directly?
Yes. Paste n rows containing only zeros and ones. The calculator checks whether every row and column has exactly one 1, then extracts the associated permutation automatically.
3) What does parity mean here?
Parity tells whether the permutation is even or odd. An even permutation uses an even number of swaps, giving determinant 1. An odd permutation uses an odd number, giving determinant -1.
4) Do fixed points affect the determinant?
Fixed points appear as 1-cycles. They matter in the cycle-count formula because all cycles are included. Still, a fixed point alone does not flip the sign unless the whole permutation structure changes.
5) Why list inversion pairs?
Inversion pairs show exactly where order is reversed. They make parity transparent and help verify the determinant step by step, especially for teaching, checking, or debugging permutation input.
6) What is minimal transposition count?
It is the fewest swaps needed to build the permutation. For size n, that count equals n minus the total number of cycles, where fixed points are counted as cycles too.
7) Can any binary matrix be used?
No. The matrix must be a true permutation matrix. That means every row and every column must contain exactly one 1. Otherwise the permutation sign formulas used here do not apply.
8) What does the graph show?
The graph plots row-to-column mapping together with cumulative inversions. It helps visualize how positions move and where disorder grows as you scan through the permutation from left to right.