We prove that for all $n\in \mathbb{ N}$, there exists a constant $C_{ n}$ such that for all $d \in \mathbb{ N}$, for every row contraction $T$ consisting of $d$ commuting $n \times n$ matrices and every polynomial $p$, the following inequality holds: \[ \|p(T)\| \le C_{ n} \sup_{ z \in \mathbb{ B}_d} |p(z)| . \] We apply this result and the considerations involved in the proof to several open problems from the pertinent literature. First, we show that Gleason's problem cannot be solved contractively in $H^\infty(\mathbb{ B}_d)$ for $d \ge 2$. Second, we prove that the multiplier algebra $\operatorname{ Mult}(\mathcal{ D}_a(\mathbb{ B}_d))$ of the weighted Dirichlet space $\mathcal{ D}_a(\mathbb{ B}_d)$ on the ball is not topologically subhomogeneous when $d \ge 2$ and $a \in (0,d)$. In fact, we determine all the bounded finite dimensional representations of the norm closed subalgebra $A(\mathcal{ D}_a(\mathbb{ B}_d))$ of $\operatorname{ Mult}(\mathcal{ D}_a(\mathbb{ B}_d))$ generated by polynomials. Lastly, we also show that there exists a uniformly bounded nc holomorphic function on the free commutative ball $\mathfrak{ C}\mathfrak{ B}_d$ that is levelwise uniformly continuous but not globally uniformly continuous.
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