Row Times Column

Matrix multiplication is the workhorse of linear algebra: each entry of the product is a dot product — row i of A dotted with column j of B. This matrix multiplication calculator performs it step by step for 2×2 and 3×3 matrices.

Why Order Matters

Unlike ordinary multiplication, A×B ≠ B×A in general. Each orientation pairs different rows with different columns, and the two products often tell completely different stories. The calculator always multiplies A on the left by B on the right.

Where It Shows Up

Composing rotations in graphics, chaining layers in neural networks, counting paths in graphs, and solving systems via Cramer's rule all reduce to matrix products. Once you can multiply, a surprising amount of applied math opens up.