We present a numerical radius bound for n×n operator matrices that improves the bound of Abu-Omar and Kittaneh (Linear Algebra Appl 468:18–26, 2015). As a significant application, we derive an estimate for the numerical radius of the Kronecker products A⊗B, where A is an n×n matrix and B is a bounded linear operator. This result refines Holbrook’s classical bound w(A⊗B)≤w(A)‖B‖ in the special case when all entries of A are non-negative. In addition, we establish spectral radius inequalities for the sums, products, and commutators of operators, improving upon the bounds of Kittaneh (Proc Am Math Soc 134:385–390, 2006) and Abu-Omar and Kittaneh (Stud Math 216(1):69–75, 2013). We further obtain an estimate for the zeros of an algebraic equation via Frobenius companion matrix, strengthening the bound of Abdurakhmanov (Mat Sb (N.S.) 131(173)(1):40–51, 126, 1986; translation in Math. USSR-Sb. 59(1):39–51, 1988). Furthermore, the Berezin radius inequalities are established, supported by several illustrative examples.