What is matrix multiplication?
Matrix multiplication combines two matrices into a single product matrix. For two 2×2 matrices A and B, the product AB is another 2×2 matrix where every entry is formed by multiplying a row of A against a column of B and summing the results — the row-by-column "dot product" rule. This calculator handles the common 2×2 case used throughout linear algebra, computer graphics, and physics.
How to use this calculator
Enter the four numbers of matrix A (A₁₁, A₁₂, A₂₁, A₂₂) and the four numbers of matrix B (B₁₁, B₁₂, B₂₁, B₂₂). Press calculate and the tool returns the full 2×2 product matrix AB. Decimals and negative numbers are fully supported.
The formula explained
The general rule is \((AB)_{ij} = \sum_{k} A_{ik} \times B_{kj}\). For 2×2 matrices this expands to four equations:
$$\left\{ \begin{aligned} C_{11} &= \text{A}_{11}\,\text{B}_{11} + \text{A}_{12}\,\text{B}_{21} \\ C_{12} &= \text{A}_{11}\,\text{B}_{12} + \text{A}_{12}\,\text{B}_{22} \\ C_{21} &= \text{A}_{21}\,\text{B}_{11} + \text{A}_{22}\,\text{B}_{21} \\ C_{22} &= \text{A}_{21}\,\text{B}_{12} + \text{A}_{22}\,\text{B}_{22} \end{aligned} \right.$$Note that matrix multiplication is generally not commutative: \(AB \neq BA\) in most cases.
Worked example
Let A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]].
$$\begin{aligned} C_{11} &= 1\times5 + 2\times7 = 5 + 14 = \mathbf{19} \\ C_{12} &= 1\times6 + 2\times8 = 6 + 16 = \mathbf{22} \\ C_{21} &= 3\times5 + 4\times7 = 15 + 28 = \mathbf{43} \\ C_{22} &= 3\times6 + 4\times8 = 18 + 32 = \mathbf{50} \end{aligned}$$So AB = [[19, 22], [43, 50]].
FAQ
Is matrix multiplication commutative? No. In general \(AB \neq BA\), so the order of the matrices matters.
What sizes can be multiplied? The number of columns of A must equal the number of rows of B. This tool focuses on 2×2 by 2×2 multiplication, which always produces a 2×2 result.
Can I use negative or decimal entries? Yes. Any real numbers are accepted, including negatives and decimals.
