Multiply Matrix A by Matrix B using row and column products
Matrix A
Matrix B
Result
Step-by-Step Derivation
Matrix Multiplication Formula
C = AB
cij = ai1 b1j + ai2 b2j + ... + ain bnj
Matrix multiplication combines rows of the first matrix with columns of the second matrix. Each result entry is a dot product, so multiplication order matters.
⚠This tool multiplies square matrices of the same selected size. For general rectangular matrices, inner dimensions must match.
How Matrix Multiplication Works
Matrix multiplication represents composition of linear transformations. It is not entrywise; every result entry depends on a row from A and a column from B.
Row by Column
Each entry is a dot product.
Order Matters
AB usually differs from BA.
Transformation Composition
Products combine linear transformations.
Square Inputs
This page supports 2x2 and 3x3 square products.
Example: For A = [[1,2],[3,4]] and B = [[5,6],[7,8]], AB = [[19,22],[43,50]].
Applications of Matrix Multiplication
Linear TransformationsComputer GraphicsMachine LearningMarkov ChainsSystems of Equations
Frequently Asked Questions
What is a matrix multiplication calculator?▼
It multiplies Matrix A by Matrix B using row-by-column dot products to produce AB.
What is the matrix multiplication formula?▼
For C = AB, each entry is cij = sum over k of aik times bkj.
How do I use this matrix product calculator?▼
Enter Matrix A and Matrix B, select 2x2 or 3x3, and calculate to get AB.
Is matrix multiplication commutative?▼
No. In general AB is not equal to BA, so the order of the two matrices matters.
Where is matrix multiplication used?▼
It is used in transformations, systems of equations, computer graphics, machine learning, and Markov chains.
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