Let $V$ be a vector space over a field $\mathbb{F}$ and for all $(i,j)\in {\mathbb{E}}_{n}^{2}$ (${\mathbb{E}}_{n}=\{1,2,...,n\}$), ${P}_{ij}:V\to V$ is a linear map. Suppose moreover that...

1) $\mathrm{\forall}({i}_{1},{j}_{1},{i}_{2},{j}_{2})\in {\mathbb{E}}_{n}^{4}$

2) $\{{v}_{j}\in V{\}}_{1\le j\le n}$ obey the following set of equations

$\mathrm{\forall}i\in {\mathbb{E}}_{n}:\phantom{\rule{thinmathspace}{0ex}}\sum _{j=1}^{n}{P}_{ij}{v}_{j}=0.$

Then $\mathrm{\forall}j\in {\mathbb{E}}_{n}$

$(detP){v}_{j}=0$

where $P$ is the $n\times n$ matrix with entries ${P}_{ij}$ and $detP$ has the same combinatorial structure as the usual determinant, yet with all multiplications replaced by map compositions.

Question: What name(s) is given to this type of result in the mathematical literature? Are there short, elegant proofs?

1) $\mathrm{\forall}({i}_{1},{j}_{1},{i}_{2},{j}_{2})\in {\mathbb{E}}_{n}^{4}$

2) $\{{v}_{j}\in V{\}}_{1\le j\le n}$ obey the following set of equations

$\mathrm{\forall}i\in {\mathbb{E}}_{n}:\phantom{\rule{thinmathspace}{0ex}}\sum _{j=1}^{n}{P}_{ij}{v}_{j}=0.$

Then $\mathrm{\forall}j\in {\mathbb{E}}_{n}$

$(detP){v}_{j}=0$

where $P$ is the $n\times n$ matrix with entries ${P}_{ij}$ and $detP$ has the same combinatorial structure as the usual determinant, yet with all multiplications replaced by map compositions.

Question: What name(s) is given to this type of result in the mathematical literature? Are there short, elegant proofs?