This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought.

Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can consider this equation over $K:=\mathrm{Frac}(R)$ and get $\lambda_1,\ldots,\lambda_n\in K$, not all zero, with the property that $$ \forall i:\quad \sum_{j=1}^n \lambda_j\cdot f_{ij} = 0 $$ Now, I can clear denominators and assume $\lambda_j\in R$. Since the $f_{ij}$ are homogeneous, I can also assume that the $\lambda_j$ are homogeneous and of minimal degree with the above property. Unfortunately, the $\lambda_j$ do not have to be constant (which is my desire), made visible by the simple counterexample $$\begin{pmatrix} x&y\\x&y\end{pmatrix} .$$ However, note that it is true for the transpose. So we similarly choose homogeneous $\mu_1,\ldots,\mu_n\in R$ which are not all zero and of minimal degree with $$ \forall j:\quad \sum_{i=1}^n \mu_i\cdot f_{ij} = 0$$

**Question:** Is it true that if the $\lambda_j$ are not all constant, then all the $\mu_i$ are constant?

I think the answer is affirmative, so my search for better counterexamples might have been half-hearted. However, so far, I cannot give a proof either. Searching the literature has yielded almost nothing, maybe I am looking in the wrong places. Your help is greatly appreciated!