Matrix Multiplication Calculator
Multiply 2x2 and 3x3 matrices with step-by-step solutions and visual representations
C = A × B, where C[i][j] = Σ A[i][k] × B[k][j]
Multiply two matrices using the dot product of rows and columns
Quick Examples
Error
Result Matrix C = A × B
Step-by-Step Calculation
=
Formula Reference
| Element | Formula |
|---|---|
| C[i][j] | Σ A[i][k] × B[k][j] |
| 2×2 Example | C[1][1] = a₁₁b₁₁ + a₁₂b₂₁ |
Important Note
Matrix multiplication is NOT commutative: A × B ≠ B × A in general
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About Matrix Multiplication Calculator
What is Matrix Multiplication?
Matrix multiplication (also called the matrix product) is a binary operation that produces a new matrix from two input matrices. Unlike regular multiplication, the order matters — A × B generally does not equal B × A.
Matrix Multiplication Condition
For two matrices A and B to be multiplied (A × B), the number of columns in A must equal the number of rows in B:
- If A is an m × n matrix
- And B is an n × p matrix
- Then C = A × B will be an m × p matrix
Matrix Multiplication Formula
Each element C[i][j] in the resulting matrix is calculated as the dot product of the i-th row of matrix A and the j-th column of matrix B:
C[i][j] = Σ A[i][k] × B[k][j] for k = 1 to n
2×2 Matrix Multiplication Example
For matrices A = [[a, b], [c, d]] and B = [[e, f], [g, h]]:
C = A × B = [[ae+bg, af+bh], [ce+dg, cf+dh]]
Step-by-Step Process
- Check dimensions: Verify that columns of A equal rows of B
- For each position (i,j): Take row i of A and column j of B
- Dot product: Multiply corresponding elements and sum them
- Place result: The sum becomes element C[i][j]
Key Properties
- Not Commutative: A × B ≠ B × A (in general)
- Associative: (A × B) × C = A × (B × C)
- Distributive: A × (B + C) = A × B + A × C
- Identity: A × I = I × A = A
Applications
- Computer Graphics: Transformations, rotations, scaling
- Machine Learning: Neural network operations
- Physics: Quantum mechanics, coordinate transformations
- Economics: Input-output models
- Engineering: Systems of equations, state-space models