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The following problem is from Golan's linear algebra book. I have been unable to make headway.

Problem: Let $n\in\mathbb{N}$ and $U$ be a non-empty finite subset of the $n\times n$ matrices over $\mathbb{C}$ which is closed under the multiplication of matrices and contains more than just the zero matrix. Show there exists a matrix $A$ in $U$ satisfying $tr(A)\in \{1,...,n\}$

EDIT: As noted in the comments, this problem is incorrect as stated. Perhaps it is correct if we allow $0$ to be in the set of desired values, or if we require the set to contain a non-singular matrix? Any help reformulating the problem would be much appreciated.

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    This is false. $U$ could contain only the zero matrix. $U$ could be empty, for that matter.2012-06-14
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    Except for Chris' comment, this question smells like you need to prove by contradiction: assume there is an $A$ with no such trace, then what is $tr(A^2)$?2012-06-14
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    there exist a matrix $A\in U\subseteq M_n(\mathbb{C})$ such that $A^k=I$, $k=|U|$2012-06-14
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    What if $U=\left\{\pmatrix{0&1\\0&0},\pmatrix{0&0\\0&0}\right\}$?2012-06-14
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    Hmm. It seems the problem is formulated incorrectly. Is there any way to salvage it?2012-06-14
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    Perhaps require that $U$ contain at least one non-singular matrix? I’ve not tried to prove the result, but this would at least rule out the obvious counterexamples.2012-06-14
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    If I remember correctly this is *not* the first time we meet an incorrect question from this Golan's book. As this book is from the Haifa University in Israel, perhaps it is compulsory (or "very recommended") to students there. I'd advice anyone studying in Israel to try the book "Algebra" by Prof. S. Amitzur, or the book with the same title by Prof.'Azriel Levi. Both exist in hebrew and are from the Hebrew University in Jerusalem. I'd rather go with the former, in spite of being a little old fashion in its notation.2012-06-14
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    @DonAntonio Be aware that Prof. Golan posts on this site. There appear to be a few incorrect question, yes, but I have found the book helpful.2012-06-14
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    @Potato Thanks for the heads-up, and mind you I find Golan's book to be pretty helpful myself, with all those hundreds of exercises. I was just trying to offer some options to eventual students who might not be aware of them in Israel, that's all.2012-06-14

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Pick any matrix $A$ in your finite set, and look at its powers. Since they are in a finite set, they repeat at some point : there exists $p,k \ge 1$ such that $A^p = A^{p+k}$. In particular, if $m$ is a multiple of $k$ and greater than $p$, then $A^{2m}=A^m$.

Thus $A^m$ is the matrix of a projection. It shouldn't be too hard now to show that its trace is an integer between $0$ and $n$ And the trace is $0$ if and only if $A$ was nilpotent in the first place. Thus you would need to accept $0$ or require that the set contains non nilpotent matrices for the statement of the exercise to be correct.