  The Jacobi-Trudi formulas imply that the minors of the banded Toeplitz
matrices can be written as certain skew Schur polynomials. In 2012,
Alexandersson expressed the corresponding skew partitions in terms of the
indices of the struck-out rows and columns. In the present paper, we develop
the same idea and obtain some new applications. First, we prove a slight
generalization and modification of Alexandersson's formula. Then, we deduce
corollaries about the cofactors and eigenvectors of banded Toeplitz matrices,
and give new simple proofs to the corresponding formulas published by Trench in
1985.
