Matrix Multiplication Calculator
Multiply matrices with step-by-step solutions and detailed explanations of the process
Matrix Multiplication
Matrix A Dimensions
Matrix B Dimensions
Results
Matrix A (2×3)
Matrix B (3×2)
Result Matrix C (2×2)
Result Properties
Step-by-Step Solution
Example Calculations
2×2 Matrix Multiplication
Matrix A:
[3 4]
Matrix B:
[7 8]
A × B:
[43 50]
2×3 × 3×2 Multiplication
Matrix A (2×3):
[4 5 6]
Matrix B (3×2):
[9 10]
[11 12]
A × B (2×2):
[139 154]
Multiplication Rules
Dimension Compatibility
A(m×n) × B(p×q) requires n = p
Columns of A = Rows of B
Result Dimensions
A(m×n) × B(n×q) = C(m×q)
Rows of A × Columns of B
Non-Commutative
A × B ≠ B × A (in general)
Order matters in matrix multiplication
Matrix Properties
Associative: (AB)C = A(BC)
Distributive: A(B + C) = AB + AC
Identity: AI = IA = A
Zero: A0 = 0A = 0
Transpose: (AB)ᵀ = BᵀAᵀ
Determinant: det(AB) = det(A)det(B)
Understanding Matrix Multiplication
What is Matrix Multiplication?
Matrix multiplication is a binary operation that combines two matrices to produce a third matrix. Unlike regular number multiplication, matrix multiplication involves computing dot products between rows and columns.
The Algorithm
- •Element calculation: C[i,j] = Σ(A[i,k] × B[k,j])
- •Row-column pairing: Row i of A with column j of B
- •Dot product: Multiply corresponding elements and sum
Mathematical Formula
For matrices A(m×n) and B(n×p):
C[i,j] = Σ(k=1 to n) A[i,k] × B[k,j]
where C is the result matrix (m×p)
Real-world Applications
- •Computer graphics and 3D transformations
- •Neural networks and machine learning
- •Solving systems of linear equations
Step 1: Check Compatibility
Verify that the number of columns in the first matrix equals the number of rows in the second matrix.
Step 2: Calculate Elements
For each position in the result, compute the dot product of the corresponding row and column.
Step 3: Form Result
Combine all calculated elements to form the final result matrix with appropriate dimensions.