## 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.

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This should be turned into a Community Wiki question. for there is no correct answer... – Mariano Suárez-Alvarez Feb 11 2010 at 23:42
Here's another example perhaps: say A and B are real n x n matrices,and A^2+B^2=AB. If AB-BA is invertible, prove that n is a multiple of 3. Are there really books that can teach you how to solve such poroblems?? – Kevin Buzzard Feb 11 2010 at 23:43
How could I make this community wiki? – zeb Feb 12 2010 at 0:10
By editing the question and checking the box which says "Community Wiki". – Mariano Suárez-Alvarez Feb 12 2010 at 0:12
@Darij: feel free to use any standard result you like from the representation theory of Lie algebras to resolve my question about real n x n matrices above! – Kevin Buzzard Feb 12 2010 at 14:56

Google will find for you V. Prasolov's Problems and Theorems in Linear Algebra, which has beautiful more or less hard problems.

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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.

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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

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Sorry to say but the Berkeley problems book is mostly spam. Most of the problems there are either well-known or boring. Actually, exam questions and problems which make one learn something are two totally different kinds of problems; the former are supposed to check one's skills, while the latter should develop them. As for the IMC, I agree - that contains many good questions, just as the [url=vjimc.osu.cz/index.php?page=history]Vojtech Jarnik[/url] and the [url=mat.itu.edu.tr/gungor/IMO/www.kalva.demon.co.uk/… contest[/url]. – darij grinberg Feb 12 2010 at 0:47
I didn't say that all problems are interesting. But I found some of the problems in the book interesting. Not all of them are standard questions, using standard techniques...! – Wanderer Feb 12 2010 at 0:49

In addition to those mentioned above, there is Linear Algebra: Challenging Problems for Students by Fuzhen Zhang

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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/

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you could also browse the linear algebra section of AoPS.

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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?