Determinant Calculator
Determinant Value:
Calculation Steps:
Understanding Determinants
What is a Determinant?
A determinant is a scalar value that can be computed from the elements of a square matrix and has several important applications in linear algebra:
- Determines if a matrix is invertible (non-zero determinant)
- Helps solve systems of linear equations
- Calculates area and volume transformations
- Finds eigenvalues of a matrix
- Determines linear independence of vectors
Properties of Determinants
- det(AB) = det(A) × det(B)
- det(Aᵀ) = det(A)
- det(A⁻¹) = 1/det(A)
- Row/column operations affect determinant
- Singular matrices have zero determinant
- Upper/lower triangular: product of diagonals
Calculation Methods
Cofactor Expansion
- Choose row/column for expansion
- Calculate minors and cofactors
- Sum products with alternating signs
- Recursive process for larger matrices
Row Reduction
- Convert to upper triangular form
- Track row operation multipliers
- Multiply diagonal elements
- Adjust sign based on swaps
Special Cases
- 2×2 matrices: ad - bc
- 3×3 matrices: Sarrus' rule
- Diagonal matrices
- Triangular matrices
Applications
- Cramer's Rule for solving equations
- Area and volume calculations
- Computer graphics transformations
- Eigenvalue problems
- Change of variables in integration
- Circuit analysis
- Quantum mechanics calculations
- Economic input-output models