https://en.wikipedia.org/wiki/Tensor_calculus is a term you're much likelier to find as the subject of a book. Matrices are just specific special cases of tensors, after all, and mathematicians like to generalize.
A good book on differential geometry will also probably start with an overview of tensor calculus.
Hubbard and Hubbard's Vector Calculus is a great introduction to multivariable calculus and linear algebra that ends with an introduction to differential forms. And it has a complete solutions manual you can purchase as well.
