  We study pairs $(A,B)$ of square matrices that are in additive (resp.
multiplicative) finite free position, that is, the characteristic polynomial
$\chi_{A+B}(x)$ (resp. $\chi_{AB}(x)$) equals the additive finite free
convolution $\chi_{A}(x) \boxplus \chi_{B}(x)$ (resp. multiplicative finite
free convolution $\chi_{A}(x) \boxtimes \chi_{B}(x)$), which equals the
expected characteristic polynomial $\mathbb{E}_U \, [ \chi_{A+U^* BU}(x) ]$
(resp. $\mathbb{E}_U \, [ \chi_{AU^* BU}(x) ]$) over the set of unitary
matrices $U$. We examine the lattice of algebraic varieties of matrices
consisting of finite free complementary pairs with respect to the additive
(resp. multiplicative) convolution. We show that these pairs include the
diagonal matrices vs. the principally balanced matrices, the upper (lower)
triangular matrices vs. the upper (lower) triangular matrices with a single
eigenvalue, and the scalar matrices vs. the set of all square matrices.
