Matrix Determinant Calculator
Calculate matrix determinants using multiple methods with detailed step-by-step solutions
Matrix Determinant Calculation
Results
Input Matrix (2×2)
Matrix Properties
Step-by-Step Solution
Geometric Interpretation
The determinant represents the area of the parallelogram formed by the matrix vectors.
Example Calculations
2×2 Matrix Example
Matrix A:
|1 4|
det(A) = 3×4 - 2×1 = 12 - 2 = 10
3×3 Matrix Example
Matrix B:
|0 4 1|
|1 2 2|
det(B) = 2(8-2) - 1(0-1) + 3(0-4) = 12 + 1 - 12 = 1
Calculation Methods
1×1 Matrix
det(A) = a₁₁
Direct value
2×2 Matrix
det(A) = ad - bc
Rule of Sarrus
3×3 Matrix
Rule of Sarrus expansion
Diagonal method
N×N Matrix
Cofactor expansion
Laplace expansion
Determinant Properties
Invertibility: det(A) ≠ 0 ⟺ A is invertible
Multiplicative: det(AB) = det(A)×det(B)
Transpose: det(Aᵀ) = det(A)
Scalar: det(kA) = kⁿ×det(A)
Inverse: det(A⁻¹) = 1/det(A)
Geometric: Represents area/volume scaling
Understanding Matrix Determinants
What is a Matrix Determinant?
The determinant of a square matrix is a scalar value that provides important information about the matrix. It tells us whether the matrix is invertible, how it scales areas or volumes, and whether it preserves or reverses orientation.
Key Applications
- •Linear systems: Determine if unique solutions exist
- •Matrix inversion: Check if matrix is invertible
- •Geometry: Calculate areas and volumes
Calculation Methods
Common Methods:
2×2: ad - bc
3×3: Rule of Sarrus or cofactor expansion
N×N: Cofactor expansion or row reduction
Geometric Meaning
- •2D: Area of parallelogram formed by column vectors
- •3D: Volume of parallelepiped formed by column vectors
- •Sign indicates orientation preservation/reversal
det(A) > 0
Matrix preserves orientation. For 2×2, parallelogram area is positive.
det(A) < 0
Matrix reverses orientation. Negative area/volume indicates flipping.
det(A) = 0
Matrix is singular (non-invertible). Vectors are linearly dependent.