In my study, I come across the following curious equality, which I do not know a proof yet (so I am asking it here).

Let $k$, $l\in \Bbb Z$ be fixed, $m$ --- the size of the below matrix $M$ --- is also fixed, and let $a_i$ be independent variables. We define $M$ by the formula: the $(i,j)$-th entry of $M$ is $a_{ik+jl}$.

Example: $k=1$, $l=2$, $m=3$, we have

$M= \left( \begin{array} {a} a_3 & a_5 & a_7 \\ a_4 & a_6 & a_8 \\ a_5 & a_7 & a_9\end{array} \right)$

Let, further, a sequence of integer numbers $\lambda=(c_1, \dots, c_m)$ be fixed. We denote by $S$ the set of all permutations of $c_1, \dots, c_m$ (the cardinality of $S$ can be less than $m!$, because some of $c_i$ can be equal).

Let $\sigma=(d_1,\dots, d_m)$ be an element of $S$. We define a matrix $M_{r,\sigma}$ ($r$ means rows) as follows: its $(i,j)$-th entry is $a_{ik+jl+d_i}$, i.e. the indices of all $a_*$ of the first row of $M$ are increased at $d_1$, the indices of all $a_*$ of the second row of $M$ are increased at $d_2$, etc.

Analogically, we define a matrix $M_{c,\sigma}$ ($c$ means columns) as follows: its $(i,j)$-th entry is $a_{ik+jl+d_j}$, i.e. the indices of all $a_*$ of the first column of $M$ are increased at $d_1$, the indices of all $a_*$ of the second column of $M$ are increased at $d_2$, etc.

Example: let $\lambda=(1,0,0)$. We have: $S$ consists of 3 elements $(1,0,0)$, $(0,1,0)$, $(0,0,1)$. For example, for $\sigma=(0,1,0)$ we have $M_{c,\sigma}=\left( \begin{array} {a} a_3 & a_6 & a_7 \\ a_4 & a_7 & a_8 \\ a_5 & a_8 & a_9\end{array} \right)$. (the indices of $a_*$ of the second column increase at 1, the indices of $a_*$ of the first and third columns remain the same).

Proposition. For all $k$, $l$, $m$, $\lambda$ we have $$\sum_{\sigma\in S} |M_{r,\sigma}| = \sum_{\sigma\in S} |M_{c,\sigma}|$$

Example. For the above $k$, $l$, $m$, $\lambda$ we have: matrices $M_{r,\sigma}$ have two equal rows for $\sigma=(1,0,0)$, $(0,1,0)$, hence the corresponding determinants are 0, and for this case the proposition becomes:

$$\left| \begin{array} {a}a_3 & a_5 & a_7 \\ a_4 & a_6 & a_8 \\ a_6 & a_8 & a_{10} \end{array} \right| = \left| \begin{array} {a} a_4 & a_5 & a_7 \\ a_5 & a_6 & a_8 \\ a_6 & a_7 & a_9\end{array} \right| +\left| \begin{array} {a} a_3 & a_6 & a_7 \\ a_4 & a_7 & a_8 \\ a_5 & a_8 & a_9\end{array} \right| + \left| \begin{array} {a} a_3 & a_5 & a_8 \\ a_4 & a_6 & a_9 \\ a_5 & a_7 & a_{10}\end{array} \right| $$

Question. 1. Is the proposition really true for all $k$, $l$, $m$, $\lambda$? Maybe they must satisfy some conditions?

- Is it already known, what is a reference?

Origin of the question: Resultantal varieties related to zeroes of L-functions of Carlitz modules http://arxiv.org/pdf/1205.2900.pdf