Quark mass matrices with full first-order perturbation

In view of current experimental constraints on the top-quark mass $m_t$ and the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements $V_{\mathrm{ij}}$, we study a modified form of the Fritzsch quark mass matrices, in which two nonzero diagonal elements for the charm and strange quarks are introduced as the additional first-order perturbative terms. In a reasonable analytical approximation, this modification can yield the upper bound of $m_t$ twice as much as that predicted by the Fritzsch Ansatz. The magnitudes of the CKM matrix elements and the parametrization-invariant measure of $\mathrm{CP}$ violation are restricted very well in terms of ratios of quark masses, and some interesting relations such as ${\left|\frac{V_{\mathrm{ub}}}{V_{\mathrm{cb}}}\right|}^2\approx \frac{m_u}{m_c}$, ${\left|\frac{V_{\mathrm{td}}}{V_{\mathrm{ts}}}\right|}^2\approx \frac{m_d}{m_s}$, and $\left|V_{\mathrm{ub}}\right|\approx \left|V_{\mathrm{ts}}\right|$ are obtained to better accuracy.