  Consider a monic linear pencil $L(x) = I - A_1x_1 - \cdots - A_gx_g$ whose
coefficients $A_j$ are $d \times d$ matrices. It is naturally evaluated at
$g$-tuples of matrices $X$ using the Kronecker tensor product, which gives rise
to its free locus $Z(L) = \{ X: \det L(X) = 0 \}$. In this article it is shown
that the algebras $A$ and $A'$ generated by the coefficients of two linear
pencils $L$ and $L'$, respectively, with equal free loci are isomorphic up to
radical. Furthermore, $Z(L) \subseteq Z(L')$ if and only if the natural map
sending the coefficients of $L'$ to the coefficients of $L$ induces a
homomorphism $A'/{\rm rad} A' \to A/{\rm rad} A$. Since linear pencils are a
key ingredient in studying noncommutative rational functions via realization
theory, the above results lead to a characterization of all noncommutative
rational functions with a given domain. Finally, a quantum version of
Kippenhahn's conjecture on linear pencils is formulated and proved: if
hermitian matrices $A_1, \dots, A_g$ generate $M_d(\mathbb{C})$ as an algebra,
then there exist hermitian matrices $X_1, \dots, X_g$ such that $\sum_i A_i
\otimes X_i$ has a simple eigenvalue.
