Linear Algebra Problems? Is there any good reference for difficult problems in linear algebra? Because I keep running into easily stated linear algebra problems that I feel I should be able to solve, but don't see any obvious approach to get started.
Here's an example of the type of problem I am thinking of: Let $A, B$ be $n\times n$ matrices, set $C = AB-BA$, prove that if $AC=CA$ then $C$ is nilpotent. (I saw this one posed on the KGS Go Server)
Ideally, such a reference would also contain challenging problems (and techniques to solve them) about orthogonal matrices, unitary matrices, positive definiteness... hopefully, all harder than the one I wrote above.
 A: Indeed, Halmos is a very good reference. You will also find some nice problems in Berkeley problems in mathematics and on the website of the International Mathematics Competition
A: In addition to those mentioned above, there is Linear Algebra: Challenging Problems for Students by Fuzhen Zhang 
A: Let me introduce you a good reference, IMAGE. At the end of IMAGE there is a section called IMAGE Problem Corner: Solutions of Old Problems and New Problems. You may enjoy solving these problems and read solutions by others.
See http://www.math.technion.ac.il/iic/IMAGE/
A: you could also browse the linear algebra section of AoPS.
A: I will take this opportunity to post  my favorite linear algebra problem.  I call it 0 not equal to 1.
Let A be an nxn 0-1 matrix with nonzero determinant.  Show that there is a 1 in every row and in every column of A, and further there is a permutation matrix P so that PA has a diagonal of all 1's.
Let B be an nxn 0-1 matrix with nonzero determinant. We cannot show that there is a 0 in every row and in every column, so assume B also has this property.  Are there nxn permutation matrices P and Q such that PBQ has all 0's on the diagonal?  If not, how small a trace can one guarantee?
Gerhard "Ask Me About System Design" Paseman, 2012.03.03 
A: If you happen to know a little bit of Italian, another good resource is Problemi risolti di algebra lineare, by Broglia, Fortuna, Luminati.
(By the way, if you have never done it, reading a math book in another language is often easier than it seems at first sight)
A: Google will find for you V. Prasolov's Problems and Theorems in Linear Algebra, which has beautiful more or less hard problems.
A: Halmos's Linear Algebra Problem Book.  It contains problems, then hints, then solutions.  There is a variety of difficulty levels, and some of the problems are very easy, but some are challenging.  The book is designed to be a supplement for learning linear algebra by problem solving, so it may not have the focus you're looking for.
