The determinant is a scalar value computed from a square matrix that encodes important properties: whether a system of equations has a unique solution, whether a matrix is invertible, and the volume scaling factor of the associated linear transformation. This calculator handles up to 4×4 matrices with step-by-step cofactor expansion.
How to Use
Select matrix size and enter elements:
- 2×2: ad - bc determinant formula
- 3×3: Cofactor expansion along first row
- 4×4: Recursive cofactor expansion
Determinant Properties
- det(AB) = det(A) × det(B)
- det(Aᵀ) = det(A)
- Swapping rows multiplies det by -1
- det = 0 means the matrix is singular (not invertible)
Applications
Determinants are used in: solving linear systems (Cramer's rule), checking matrix invertibility, computing cross products, calculating area/volume of parallelograms/parallelepipeds, and eigenvalue problems.
Step-by-Step Instructions
- 1Select the matrix size (2×2, 3×3, or 4×4).
- 2Enter the matrix elements.
- 3View the determinant value.
- 4Check invertibility and matrix properties.
- 5Study the cofactor expansion steps.