Matrix Multiplication Calculator
Multiply 2D matrices instantly. Calculate the product of two matrices with customizable dimensions, real-time validation, and step-by-step linear algebra logic.
Matrix Multiplier
Define dimensions and enter values
Note: Columns of A must equal Rows of B for multiplication to be valid.
Multiplication Result
Product matrix and calculation details
Enter your matrix dimensions and values, then click Calculate Product to see the resulting matrix.
Properties of Matrix Multiplication
Key mathematical rules and properties that govern how matrices interact.
| Property | Rule / Formula | Description |
|---|---|---|
| Dimension Rule | A(m×n) × B(n×p) = C(m×p) | Inner dimensions must match; result takes outer dimensions. |
| Non-Commutative | AB ≠ BA | Order matters; reversing matrices usually changes the result. |
| Associative | (AB)C = A(BC) | Grouping does not affect the final product. |
| Distributive | A(B + C) = AB + AC | Multiplication distributes over matrix addition. |
| Identity Matrix | AI = IA = A | Multiplying by the identity matrix leaves the original unchanged. |
| Zero Matrix | A0 = 0A = 0 | Multiplying by a zero matrix results in a zero matrix. |
Matrix Multiplication FAQ
Everything you need to know about multiplying matrices and linear algebra basics.
Matrix multiplication is a binary operation that takes a pair of matrices and produces another matrix. It is a fundamental operation in linear algebra, used extensively in physics, computer graphics, machine learning, and engineering to represent linear transformations and solve systems of linear equations.
You can multiply two matrices only if they are compatible. Specifically, the number of columns in the first matrix (Matrix A) must exactly equal the number of rows in the second matrix (Matrix B). If A is an m × n matrix and B is an n × p matrix, their product AB will be an m × p matrix.
No, matrix multiplication is generally not commutative. This means that for two matrices A and B, the product AB is usually not equal to BA. The order in which you multiply matrices matters, which is a key difference from standard multiplication of real numbers.
Each element in the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and the corresponding column from the second matrix. You multiply matching entries and sum them up. For example, the element in row i and column j of the product is the sum of A[i,k] * B[k,j] for all valid k.
Matrix multiplication is used in 3D computer graphics to rotate, scale, and translate objects. In machine learning, it is the core operation behind neural networks (forward propagation). It is also used in quantum mechanics, economics for input-output models, and solving systems of linear equations in engineering.