Books on linear algebra more focused towards matrices and determinants rather than vector spaces In my syllabus of a competitive exam, we have matrices and determinants and solving linear equations with them instead of linear algebra but during examinations a lot of the times questions are derived from linear algebra and presented such that they can be tackled by matrices and determinants but are tedious. Eg, they will give a 2×2 matrix and then give 4 polynomial as options , where one of them would be a characteristic polynomial. So anyone who know about Cayley-Hamilton Theorem, will easily solve this quickly. 
So are there any books  available that have a lot of these properties like properties of eigenvectors, idempotent, nilpotent matrices,symmetric, skew symmetric etc. Most of the popular books revolve around teaching the vector spaces and that part really well. 
I have studied linear algebra from MIT OCW by Gilbert Strang and have partially read his book on the subject too, so even if the book has some parts from vector spaces (which it probably will) , I believe, I will probably be able to comprehend what the author is trying to convey. 
Thanks 
 A: I strongly recommend the book Matrix Theory: Basic Results and Techniques by Fuzhen Zhang. This is the book that had all of the algebraic tricks I had to learn on my own for matrix calculations. The writing is clear and the focus is on teaching both important results and algebraic and analytic techniques used to derive them. A particular standout is the book's four proofs of the fact that $AB$ and $BA$ have the same nonzero eigenvalues. There are sections on most important classes of matrices (symmetric, orthogonal, idempotent, nilpotent,...), and the book covers everything from the basics to some pretty serious matrix analysis content at the end. I truly feel that this is an underappreciated gem and should be a go-to text for anyone who frequently encounters matrices.
A: The book Matrix Mathematics: Theory, Facts, and Formulas by Dennis S. Bernstein can be useful. For example, from the section Facts on Nilpotent Matrices:

If B is nilpotent and $AB = BA$, then $\det(A + B) = \det A$.

A: Matrix Algebra by Abadir & Magnus that, in the first ~200 pages, gives most of core results in linear algebra mostly using matrix calculation techniques but not vector space theory. The latter part goes into some matrix theory. What is really great of this book is that the calculations are often enlightening rather than tedious! (The latter is true for many matrix algebra books.)
Don’t be cheated by the series title. It is written in the format of exercises, but contains all the important results and complete proofs (as solutions). It is in the series because it is needed by econometrics, but it is not on econometrics. I highly recommend it to people in statistics, optimization and machine learning, etc.
Edit: I have took a brief look at the book recommended by @eepperly16. It is a great book on matrix theory with a focus on calculation techniques. However, it does not derive or enlighten linear algebra results that are usually given by vector space theory, but rather assume them as prerequisites.
Edit2: After a second reading of the question, it seems, particularly from the second paragraph, a matrix theory book is asked. If so, Zhang’s book is indeed a better choice. But I hope others attracted in by the title of the question would find Abadir & Magnus more interesting.
