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I already posted this question on Mathematics StackExchange. A user there suggested that I rather post it on mathoverflow, since it is a research question. So here it is. Let $m\geq2$, $n\geq1$ be ...

We have $\det M=(a+b)(c+d)$ where $M=\begin{bmatrix} a& 0& -1& 0\\ 0& c& 0& -1\\ b& 0& 1& 0\\ 0& d& 0& 1 \end{bmatrix}$ and $\det M'=(a'+b')(c'+d')$...

I have tried many ways to prove the following equation, such as the method of induction and expanding all the terms in the summation,but things got more complicated.I could not find an appropriate ...

Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$. Does there exist a ...

For $1 \le i, j \le k$, consider $\rho_{ij}$ which are equal to either zero or one such that $\rho_{ii}=1$ and $\rho_{ij}=0$ if and only if $\rho_{ji}=0$. How to find the zeros of the determinant of ...