Consider the $3 \times 3$ matrix polynomial

$$M(x)=\left(\begin{array}{ccc}Ax&Bx&Cx\\1&1&1\end{array}\right)$$

where $A, B, C$ are $2 \times 2$ real matrices. Assume that $\mbox{rank} (M(x)) \leq 2$ for all $x$ in a neighborhood of zero. Then, by a direct calculation, we have that the set $\{A,B,C\}$ is linearly dependent.

Is this a manifestation of a more general phenomenon, or pure luck?

Edit: The result fails without symmetry of the matrices, and holds for $m$ symmetric matrices $A_i\in\mathbb{R}^{n\times n}, i=1,\dots,m$. That is, $\mbox{rank}(M(x))\leq 2, \forall x\in\mathbb{R}^n$ implies $\mbox{rank}\{A_1,\dots,A_m\}\leq2$.

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