✖️ Matrix Multiplication Calculator
Multiply two matrices and compute A × B
How to Use This Calculator
Set Matrix Dimensions
A: m×n and B: n×p. Columns of A must equal rows of B.
Enter Matrix Elements
Input all elements of matrices A and B.
Calculate
Click to compute the product matrix A × B.
View Result
The result has dimensions m×p (rows of A × columns of B).
Formula
(A × B)ᵢⱼ = Σₖ AᵢₖBₖⱼ
Element (i,j) is dot product of row i of A and column j of B
Definition:
Matrix multiplication computes element (i,j) as the dot product of row i of A and column j of B.
Example for 2×2 matrices:
[a₁₁ a₁₂][b₁₁ b₁₂] [a₁₁b₁₁+a₁₂b₂₁ a₁₁b₁₂+a₁₂b₂₂]
[a₂₁ a₂₂][b₂₁ b₂₂] = [a₂₁b₁₁+a₂₂b₂₁ a₂₁b₁₂+a₂₂b₂₂]
Requirements:
- A: m×n matrix, B: n×p matrix
- Number of columns in A = number of rows in B
- Result is m×p matrix
Properties:
- Not commutative: A × B ≠ B × A in general
- Associative: (A × B) × C = A × (B × C)
- Distributive: A × (B + C) = A × B + A × C
About Matrix Multiplication Calculator
The Matrix Multiplication Calculator computes the product of two matrices. Matrix multiplication is fundamental in linear algebra, used for transformations, solving systems, and matrix operations.
When to Use This Calculator
- Linear Transformations: Composing transformations
- Systems of Equations: Matrix form representation
- Graphics: Coordinate transformations
- Data Analysis: Weighted combinations
- Physics: Quantum mechanics, rotations
Why Use Our Calculator?
- ✅ Flexible Sizes: Supports various matrix dimensions
- ✅ Dimension Validation: Ensures compatible matrices
- ✅ Clear Display: Shows input and output matrices
- ✅ Educational: Helps understand matrix multiplication
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Dot Product: Each element is a dot product of row and column
- Compatibility: Inner dimensions must match
- Non-Commutative: A × B ≠ B × A generally
- Order Matters: (A × B) × C = A × (B × C)
- Identity: I × A = A × I = A
Frequently Asked Questions
Can I multiply any two matrices?
No, the number of columns in the first matrix must equal the number of rows in the second. For A: m×n and B: p×q, you can only compute A × B if n = p.
Is matrix multiplication commutative?
Generally no! A × B ≠ B × A. Even for square matrices, multiplication is not commutative, unless the matrices commute (e.g., both are identity or diagonal).
What if I try to compute A × B but dimensions don't match?
The calculator will display an error. You may need to rearrange matrices or compute B × A instead if dimensions permit.
How is this different from element-wise multiplication?
Matrix multiplication uses dot products (Σ product). Element-wise (Hadamard) multiplication multiplies corresponding elements: (A ⊙ B)ᵢⱼ = AᵢⱼBᵢⱼ. They're completely different!
Can I multiply more than two matrices?
Yes, use associativity: A × B × C = (A × B) × C or A × (B × C). Order within groups matters, but grouping can be rearranged.