Matrix Multiplication Calculator

Matrix A
Rows:
Columns:
×
Matrix B
Rows:
Columns:

Result Matrix:

Calculation Steps:

Understanding Matrix Multiplication

What is Matrix Multiplication?

Matrix multiplication is a fundamental operation in linear algebra where two matrices are combined to create a new matrix. For matrices A(m×n) and B(n×p):

  • The number of columns in A must equal the number of rows in B
  • The resulting matrix C will have dimensions m×p
  • Each element c[i,j] is the dot product of row i from A and column j from B
  • The operation is not commutative (AB ≠ BA in general)

Properties of Matrix Multiplication

  • Associativity: (AB)C = A(BC)
  • Distributivity: A(B+C) = AB + AC
  • Non-commutativity: AB ≠ BA
  • Identity matrix: AI = IA = A
  • Zero matrix: A0 = 0A = 0
  • Transpose: (AB)ᵀ = BᵀAᵀ

Applications

Computer Graphics

  • 3D transformations
  • Rotation matrices
  • Scaling operations
  • Perspective projections
  • Animation transformations

Data Science

  • Neural networks
  • Principal component analysis
  • Dimensionality reduction
  • Feature extraction
  • Covariance matrices

Physics

  • Quantum mechanics
  • Rigid body dynamics
  • Stress tensors
  • Inertia calculations
  • Wave propagation

Computational Aspects

  • Time complexity: O(n³) for naive algorithm
  • Strassen's algorithm
  • Parallel computation
  • Block matrix multiplication
  • Numerical stability
  • Cache optimization