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Let $S_m(q)$ denote the space of all $m\times m$ symmetric matrices over the finite field $\mathbb{F}_q$ of size $q$. What is the number of matrices $A=(a_{ij})\in S_m(q)$ of rank at most $3$ and $a_{11} + a_{22}\neq 0$.

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  • $\begingroup$ "size $q$" should be "size $m$"? $\endgroup$ – Gerry Myerson Feb 5 '18 at 22:41
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    $\begingroup$ Here is a paper that counts for each fixed rank $r$ how many symmetric matrices there are- this ignores your last condition math.clemson.edu/~kevja/REU/2004/SymmetricRankRMatrices.pdf $\endgroup$ – Vlad Matei Feb 6 '18 at 0:27
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    $\begingroup$ @ Gerry Myerson >Sorry for the confusion. I meant there the size of the field. But I will change it. $\endgroup$ – Singh Feb 6 '18 at 9:52
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    $\begingroup$ @VladMatei Here is an extra condition on the diagonal elements of those matrices, i.e. $a_{11} +a_{22}\neq 0$ $\endgroup$ – Singh Feb 6 '18 at 9:57
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    $\begingroup$ Have you looked at small values of $m$? If a pattern shows itself, that would be encouraging. Nothing about the questions suggests to me it will have a nice answer, but I haven't really got much intuition for the question. $\endgroup$ – Hugh Thomas supports Monica Feb 6 '18 at 12:17

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