Determinant Calculator
Laplace Expansion Calculator
Result
Determinant:
Steps
About Determinant Calculator
Determinant Calculator computes the determinant of a square matrix using the Laplace expansion method, also known as cofactor expansion. This method is widely used in linear algebra to break down the determinant calculation into smaller submatrices.
How to Use
- Select the matrix size (2×2 to 8×8) from the dropdown menu.
- Enter the matrix elements using the keypad or your keyboard. Use the arrow buttons to navigate between input fields.
- Click the “Calculate” button to compute the determinant using Laplace expansion.
- The result will be displayed below, along with detailed step-by-step calculations.
- Use the “Copy Result” button to copy the determinant value and steps.
- Share your result on Twitter, WhatsApp, or Facebook, or copy an embed code to include the calculator or result on your website using the share buttons.
Formulas
The Laplace expansion along the i-th row of a matrix B is given by:
det(B) = Σ (-1)^(i+j) * b_(i,j) * m_(i,j) for j from 1 to n
Similarly, along the j-th column:
det(B) = Σ (-1)^(i+j) * b_(i,j) * m_(i,j) for i from 1 to n
Where:
- b_(i,j) is the element at row i and column j.
- m_(i,j) is the minor, the determinant of the submatrix obtained by removing row i and column j.
Important Notes
- The calculator supports matrices up to 8×8 to balance usability and computational feasibility.
- Laplace expansion has O(n!) complexity, making it computationally intensive for large matrices like 8×8. For larger matrices, consider methods like LU decomposition (O(n³) complexity).
- Ensure all inputs are valid numbers. Invalid inputs will result in an error message.
- Calculations for larger matrices may take longer due to the recursive nature of Laplace expansion.
- Clipboard access requires a secure context (HTTPS). If copying fails, try accessing the calculator over HTTPS or copy manually.