Free Matrix Calculator - Complete Linear Algebra Tool

Our comprehensive matrix calculator performs all essential linear algebra operations online. Calculate matrix addition, subtraction, multiplication, determinants, inverses, and transposes with ease. Perfect for students, engineers, and professionals working with linear algebra problems.

Whether you're solving systems of linear equations, performing transformations, or working on complex mathematical problems, our matrix calculator provides accurate results with clear, step-by-step explanations. Support for matrices up to 5×5 with instant calculations and detailed solutions.

Matrix Operations Calculator

Matrix A

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Matrix B

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Understanding Matrix Operations

Basic Operations

Matrix Addition

Add corresponding elements: (A + B)ᵢⱼ = Aᵢⱼ + Bᵢⱼ

[1 2] + [5 6] = [6 8]
[3 4] [7 8] [10 12]

Matrix Multiplication

Dot product of rows and columns: (AB)ᵢⱼ = Σ Aᵢₖ × Bₖⱼ

[1 2] × [5 6] = [19 22]
[3 4] [7 8] [43 50]

Advanced Operations

Determinant

For 2×2: det(A) = ad - bc

det([a b]) = ad - bc
    ([c d])

Matrix Inverse

A⁻¹ exists if det(A) ≠ 0. AA⁻¹ = I

A⁻¹ = (1/det(A)) × adj(A)

Real-World Applications of Matrix Operations

Engineering & Physics

Structural analysis, circuit theory, quantum mechanics, and transformation matrices in computer graphics and robotics.

Computer Science

Machine learning algorithms, image processing, database operations, and graph theory applications.

Economics & Finance

Portfolio optimization, input-output models, risk analysis, and economic forecasting models.

Tips for Matrix Calculations

Best Practices

  • 1
    Always verify matrix dimensions before performing operations
  • 2
    Check for singular matrices before calculating inverses
  • 3
    Use the copy function to transfer results to other applications
  • 4
    Start with smaller matrices to understand operations

Common Mistakes to Avoid

  • Attempting to add matrices with different dimensions
  • Confusing row-column multiplication rules
  • Trying to find inverse of singular matrices
  • Forgetting that matrix multiplication is not commutative