4×4 Determinant Input Form
Enter the sixteen matrix values, choose a row or column, and calculate the determinant by Laplace expansion.
Formula Used
For a 4×4 matrix A, Laplace expansion writes the determinant as a sum of entry-by-cofactor products.
Row expansion: det(A) = ar1Cr1 + ar2Cr2 + ar3Cr3 + ar4Cr4
Column expansion: det(A) = a1cC1c + a2cC2c + a3cC3c + a4cC4c
Cofactor rule: Cij = (-1)i+j × det(Mij)
The calculator removes one row and one column to create each 3×3 minor matrix Mij. It then finds the signed cofactor, multiplies by the selected entry, and sums the four contributions. Expanding through a row or column with more zeros usually reduces manual work.
How to Use This Calculator
- Enter all sixteen values of the 4×4 matrix.
- Choose whether you want a row expansion or a column expansion.
- Select the exact row or column number from 1 to 4.
- Set the display precision for rounded output, if needed.
- Click Calculate Determinant to view the determinant, cofactors, term contributions, graph, and export buttons.
- Use the example button to load a ready matrix and compare its expansion details.
Example Data Table
| Row | Column 1 | Column 2 | Column 3 | Column 4 |
|---|---|---|---|---|
| 1 | 2 | 1 | 0 | 3 |
| 2 | -1 | 4 | 2 | 1 |
| 3 | 3 | 0 | 1 | -2 |
| 4 | 2 | 5 | -1 | 4 |
| Expansion Choice | Row 1 | |||
| Determinant | -160 | |||
FAQs
1. What does the determinant tell me?
The determinant shows whether a square matrix is singular or invertible. A zero determinant means the matrix has no inverse and its rows or columns are linearly dependent.
2. Why use Laplace expansion for a 4×4 matrix?
Laplace expansion is useful for learning structure, cofactors, and minors. It is slower than elimination for large systems, but it clearly shows how each selected row or column contributes to the determinant.
3. Does the chosen row or column change the answer?
No. Every valid row or column expansion must produce the same determinant. The selected expansion only changes the intermediate cofactors and the amount of arithmetic work shown.
4. Which row or column should I expand?
Choose the row or column with the most zeros or simpler values. That reduces nonzero terms, makes the cofactor calculations easier, and lowers the chance of arithmetic mistakes.
5. Can I enter decimals or negative values?
Yes. The calculator accepts integers, decimals, and negative numbers. Adjust the display precision if you want the result and cofactors rounded to a specific number of decimal places.
6. What is a minor matrix?
A minor matrix is the 3×3 matrix left after removing one row and one column from the original 4×4 matrix. Its determinant is used to build the cofactor.
7. What is the sign pattern for cofactors?
Cofactor signs follow a checkerboard pattern: +, −, +, − on the first row, then alternating on each next row. The calculator applies this using the factor (−1)i+j.
8. What do the CSV and PDF downloads include?
They include the current matrix values, selected expansion line, determinant, matrix status, and each term’s entry, minor determinant, cofactor, and signed contribution for documentation or review.