We study a simple inductive data type for representing correct-by-construction matrices. Despite its simplicity, it can be used to implement matrix-manipulation algorithms efficiently and safely, performing in some cases faster than existing alternatives even though the algorithms are written in a direct and purely functional style. A rich collection of laws makes it possible to derive and optimise these algorithms using equational reasoning, avoiding the notorious off-by-one indexing errors when fiddling with matrix dimensions. We demonstrate the usefulness of the data type on several examples, and highlight connections to related topics in category theory.