Matrix Multiplication Calculator
Multiply two matrices using the dot product method.
Calculate Matrix Multiplication
Matrix Multiplication Examples
2×2 Matrices
A: 1,2;3,4
B: 5,6;7,8
Result: 19,22;43,50
Identity Matrix
A: 1,2;3,4
B: 1,0;0,1
Result: 1,2;3,4 (same as A)
2×3 × 3×2
A: 1,2,3;4,5,6
B: 1,2;3,4;5,6
Result: 22,28;49,64
How to Calculate Matrix Multiplication
Requirements
For matrices A (m×n) and B (n×p), the product AB exists and has size m×p.
The number of columns in A must equal the number of rows in B.
Process
For each element (i,j) in the result matrix:
(AB)[i,j] = Σ(A[i,k] × B[k,j])
Where the sum is taken over k from 1 to n.
Properties
- Matrix multiplication is associative: (AB)C = A(BC)
- Matrix multiplication is distributive: A(B+C) = AB + AC
- Matrix multiplication is NOT commutative: AB ≠ BA (in general)
- AI = IA = A (where I is identity matrix)
Special Cases
- Identity matrix: AI = A
- Zero matrix: A0 = 0
- Diagonal matrices: Multiply element-wise
- Scalar multiplication: kA = [k×a[i,j]]