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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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    $\begingroup$ 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?? $\endgroup$
    – Kevin Buzzard
    Commented Feb 11, 2010 at 23:43
  • $\begingroup$ "Are there really books that can teach you how to solve such problems??" Books about the representation theory of Lie algebras perhaps. At least this is where most of such problems come from. Not that I would know the techniques for solving them, however. Except the standard tricks (a matrix $A$ over a field of characteristic zero is nilpotent if and only if every k = 1, 2, 3, ... satisfies Tr (A^k) = 0, etc.) $\endgroup$
    – darij grinberg
    Commented Feb 12, 2010 at 0:42
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    $\begingroup$ @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! $\endgroup$
    – Kevin Buzzard
    Commented Feb 12, 2010 at 14:56
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    $\begingroup$ I should now perhaps add that the two spammy answers that motivated me to write above comment are now gone. Of course I have no problem with the now recent ones. $\endgroup$
    – user9072
    Commented Mar 4, 2012 at 0:02
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    $\begingroup$ @Kevin, see math.stackexchange.com/questions/299651/… for a somewhat systematic approach. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Jan 17, 2018 at 21:58

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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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    $\begingroup$ 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]. $\endgroup$
    – darij grinberg
    Commented Feb 12, 2010 at 0:47
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    $\begingroup$ 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...! $\endgroup$
    – Wanderer
    Commented Feb 12, 2010 at 0:49
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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?

Gerhard "Ask Me About System Design" Paseman, 2012.03.03

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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)

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