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The following determinant arises in a combinatorial enumeration problem. I wonder if anyone has seen it before in any context or knows how to evaluate it. I tried computing it for small $n$ but didn't …

Let $A_1,A_2$ be $n\times n$ symmetric matrices over $\{0,1\}$, and let $p_1, p_2$ be their respective characteristic polynomials over the rationals. If $p_1 \ne p_2$, then there is some positive inte …

You have a class of graphs closed under some notion of isomorphism, and a function $f$ defined on that class that is invariant under the same notion of isomorphism. For a graph $g$ let $a(g)$ be the o …

Let $L_i$ and $R_i$ be the $i$-th rows of the left and right matrices. Then $$\begin{align} R_n &= L_n \\ R_{n-1} &= L_n + L_{n-1} \\ R_{n-2} &= L_n + 2L_{n-1} + L_{n-2} \\ R_{n-3} …

The eigenvalues of $A+\lambda$ are $\{\mu_j+\lambda\}$ which are positive by assumption. So $$\frac{d}{d\lambda} \log\det(A+\lambda) = \frac{d}{d\lambda} \sum_j \log (\lambda+\mu_j) = \sum_j (\lambd …

$$B=\left[\matrix{1&1&1&1\\1&1&-1&-1\\-1&-1&-1&1\\-1&1&-1&1}\right]$$ and probably many other solutions. I'm also voting to close because you didn't pose a research-level problem. If you have an int …

If there is a rational nonzero solution, there is an integer nonzero solution by multiplying up. At least one of the integers can be assumed odd by dividing out a common power of two. The determinant …