Abstract: Motivated by the need to formalise generation of fast running code for linear algebra applications, we show how an index-free, calculational approach  to matrix algebra can be developed by regarding matrices as morphisms of  a category with biproducts and its self-adjoint functors. This shift, in the traditional view of matrices as indexed structures to a type-level perspective analogous to that of the pointfree algebra of programming, makes it easy to calculate algorithms implementing matrix multiplication, the kernel operation of matrix algebra, ranging from its divide-and-conquer version to the conventional, iterative one and several vectorised variations.