The generalized linear operator on a vector space over a commutative semiring which has zero and identity elements, idempotent addition, and invertible multiplication is considered. Some useful inequalities for the norm, trace, and eigenvalue of matrix are established. Based on the inequalities, simple proofs are suggested for the convergence theorems for the growth rate of the norm and trace of powers of the operator to converge to its spectral radius as the exponent tends to infinity. It is shown that the general expression for the spectral radius can be obtained as a consequence of the theorems.