2x2 Matrix Multiplication Calculator

About this calculator

Multiplying two 2×2 matrices follows the row-by-column rule: each entry of the result is the sum of products formed by pairing a row from the first matrix with a column from the second. Unlike ordinary number multiplication, matrix multiplication is not commutative — A×B generally gives a different result than B×A — because the rule pairs rows against columns in a fixed order that depends on which matrix comes first.

This operation is the mechanism behind composing linear transformations: multiplying a 2×2 rotation matrix by a scaling matrix, for example, produces a single matrix that rotates and scales in one step, which is exactly how 2D computer graphics, robotics kinematics and physics problems chain multiple transformations together efficiently. Small matrices like the 2×2 Pauli matrices also appear directly in quantum mechanics to represent spin operators.

Enter the entries of two 2×2 matrices and this calculator multiplies them using the row-by-column rule to give you the resulting matrix.