Exponents of $2$-multiarrangements and Wakefield--Yuzvinsky matrices

In the theory of hyperplane arrangements, M. Wakefield and S. Yuzvinsky utilized a square matrix in their research on the exponents of $2$-dimensional multiarrangements. Using such a matrix, they showed that the exponents of $2$-dimensional multiarrangements are as close as possible in general position for any fixed balanced multiplicity. In this article, we introduce a matrix similar to that of Wakefield and Yuzvinsky and explore further applications to the exponents. In fact, the exponents of $2$-dimensional multiarrangements are determined by whether the corresponding matrices have full rank. As one of our main results, we introduce a new class of $2$-dimensional arrangements for which the exponents are as close as possible for any balanced multiplicities, except for the constant one multiplicity. We also proceed with the classification of $B_2$-exponents, and we provide an alternative proof for some known results on the exponents.