Matrix Multiplication Calculator (2x2)

This calculator multiplies two 2 by 2 matrices and shows the resulting product matrix. Matrix multiplication is one of the central operations in linear algebra, and it works quite differently from multiplying ordinary numbers. To find each entry of the product, you take a row of the first matrix and a column of the second, multiply their corresponding elements, and add the results, a row-by-column dot product. This rule, while a little involved, is what makes matrices so powerful: multiplying matrices composes transformations, so a single product can represent rotating then scaling then translating in computer graphics, or chaining the steps of a process. Matrix multiplication underlies graphics and game engines, the linear layers of neural networks, systems of equations, and much of physics and engineering. This tool removes the arithmetic. You enter the four entries of the first matrix and the four of the second, and the calculator returns the product matrix and each of its four entries individually. The results update as you type, so you can explore how the product changes. Use it for linear algebra homework, for graphics and transformation work, or to check a hand calculation. A crucial property the calculator illustrates is that matrix multiplication does not commute: multiplying A by B generally gives a different result from B by A, unlike ordinary numbers, because the order in which transformations are applied matters. Each entry of the product comes from one row of the first matrix and one column of the second: the top-left entry uses the first row and first column, the top-right uses the first row and second column, and so on. The calculations are exact for the values you enter.

Matrix A = [a b; c d], B = [e f; g h]. Each product entry is a row of A times a column of B. Matrix multiplication does not commute: A x B is generally not B x A.