Matrix Determinant Calculator
Calculate matrix determinants for 2×2, 3×3, and 4×4 matrices with comprehensive analysis. Our advanced linear algebra calculator provides step-by-step solutions, cofactor matrices, inverse matrices, and complete matrix properties for educational and professional use.
Last updated: December 15, 2024
Need a custom calculator for your business? Get a Quote
Quick Example Result
For the 2×2 matrix [[2,3],[1,4]]:
Determinant
5
Trace
6
Type
non-singular matrix
Invertible
Yes
How This Calculator Works
Our matrix determinant calculator uses advanced linear algebra algorithms to compute determinants efficiently and accurately. It employs different methods based on matrix size: direct calculation for 2×2 matrices, cofactor expansion for 3×3 matrices, and recursive cofactor expansion for larger matrices, providing comprehensive analysis including matrix properties and inverse calculations.
Determinant Calculation Methods
2×2 Matrix Formula:
det([[a,b],[c,d]]) = ad - bcDirect calculation using the simple cross-multiplication method
3×3 Matrix (Sarrus Rule):
det = a(ei-fh) - b(di-fg) + c(dh-eg)Cofactor expansion along the first row for 3×3 matrices
Larger Matrices (Cofactor Expansion):
det(A) = Σ(aᵢⱼ × Cᵢⱼ)Recursive cofactor expansion where Cᵢⱼ = (-1)^(i+j) × Mᵢⱼ
Matrix Inverse Formula:
A⁻¹ = (1/det(A)) × adj(A)Where adj(A) is the adjugate matrix (transpose of cofactor matrix)
These methods ensure accurate computation of determinants and related matrix properties, providing both educational insight and practical utility for linear algebra applications.
Interactive matrix grid showing element positions and calculation patterns
Matrix Properties and Analysis
Beyond calculating the determinant, our calculator provides comprehensive matrix analysis including matrix classification, invertibility, trace calculation, and when applicable, the complete inverse matrix with cofactor and adjugate matrices for educational purposes.
- Singular matrices: Determinant = 0, not invertible, represent collapsed transformations
- Non-singular matrices: Determinant ≠ 0, invertible, preserve dimensional properties
- Matrix trace: Sum of diagonal elements, important for eigenvalue analysis
- Matrix rank: Maximum number of linearly independent rows or columns
Linear Algebra Applications & Standards
- International Linear Algebra Society (ILAS) - Standards for Linear Algebra Computation and EducationProfessional standards for matrix computations and numerical methods
- Society for Industrial and Applied Mathematics (SIAM) - Computational Linear Algebra GuidelinesIndustry standards for numerical linear algebra applications
- IEEE Computer Society - Standards for Matrix Computation AlgorithmsTechnical standards for computational accuracy and efficiency
Need help with other math calculations? Check out our Law of Cosines calculator and quadratic formula calculator.
Get Custom Calculator for Your BusinessExample Calculation
Given Matrix:
Calculation Steps:
- Expand along first row: det = 1×C₁₁ - 2×C₁₂ + 3×C₁₃
- C₁₁ = det([[5,6],[8,9]]) = 5×9 - 6×8 = -3
- C₁₂ = det([[4,6],[7,9]]) = 4×9 - 6×7 = -6
- C₁₃ = det([[4,5],[7,8]]) = 4×8 - 5×7 = -3
- det = 1×(-3) - 2×(-6) + 3×(-3) = -3 + 12 - 9 = 0
Result: Determinant = 0, indicating this is a singular matrix (not invertible)
This matrix has linearly dependent rows (each row is an arithmetic progression), which results in a zero determinant. Such matrices cannot be inverted and represent transformations that collapse 3D space into a lower dimension.
Frequently Asked Questions
Found This Calculator Helpful?
Share it with others who might need to calculate matrix determinants and analyze linear transformations
Suggested hashtags: #Matrix #Determinant #LinearAlgebra #Math #Calculator