There are a lot of linear algebra-matrix theory books around, and for my next course I picked Shilov's book. (It is advanced undergraduate/beginning graduate level.) So what is special about this it? It has a good mix of theory and computation.
It has many exercises, and they help in the learning, I think, better than some of the books I looked at as well. Some of them have hints, and some answers, listed in the back. Processing matrices is a visual thing. Shilov's book includes lovely pictures of matrices, and their block components. This is a big help in teaching matrix products, inverses, determinants, diagonalizations, not to mention applications to geometry, conical sections, projective space, and tangent planes. The beautiful pictures are many, are superb, are artistically done and include graphical illustrations, figures, surfaces and more. They may not be created with the latest software, and yet they are timeless, and better IMO than the glossy ones found in newer editions.
Another thing I like about Shilov's book is its use of multilinear algebra. It is motivated by differential geometry of course, but it is great for applications to for example invariant theory.
There is a number of great alternative choices, some priced in the $ 100 range; and yet I found that Shilov ($ 10) has an edge; even forgetting the price. Sample of topics: Standard fare on linear algebra and matrices, systems of equations etc; vector spaces and their duals, inner product, linear transformations and their adjoints; canonical form, quadratic forms & extrema, unitary space, Jordan forms, and a nice selection of timeless applications. Review by Palle Jorgensen, December 2008.