Matrix Multiplication Calculator
Not Just Multiplying Matching Cells
Matrix multiplication trips people up precisely because it isn't element-by-element like addition or subtraction — each entry in the result is a full dot product between a row of the first matrix and a column of the second. That single difference is what lets matrices represent chained transformations, not just paired data.
The Formula
C = A × B, where Cij = Σ(Aik × Bkj)
Worked Example
For A = [1, 2; 3, 4] and B = [5, 6; 7, 8]:
| Entry | Calculation | Result |
|---|---|---|
| C11 | (1×5) + (2×7) | 19 |
| C12 | (1×6) + (2×8) | 22 |
| C21 | (3×5) + (4×7) | 43 |
| C22 | (3×6) + (4×8) | 50 |
Result matrix: [19, 22; 43, 50].
Where This Matters
- Chaining transformations — combining a rotation and a scaling into a single transformation matrix in computer graphics or robotics.
- Neural networks — every layer of a neural network multiplies an input matrix by a weight matrix as its core operation.
- Markov chains — multiplying a state vector by a transition matrix predicts the next state in probabilistic models.
Note: Matrix multiplication is not commutative — A × B generally does not equal B × A, even when both products are defined.
How to Use This Calculator
- Choose the matrix size: 2 x 2 Matrices or 3 x 3 Matrices.
- Enter each cell of Matrix A.
- Enter each cell of Matrix B.
- Select Calculate to get the product matrix.
Related Calculations
Undo a transformation with the Matrix Inverse Calculator, or combine matrices additively instead with the Matrix Addition Calculator.