Bounds on quark mass matrices elements due to measured properties of the mixing matrix and present values of the quark masses

We obtain constraints on possible structures of mass matrices in the quark sector by using as experimental restrictions the determined values of the quark masses at the $M_Z$ energy scale, the magnitudes of the quark mixing matrix elements $V_{\mathrm{u}\mathrm{d}}$, $V_{\mathrm{u}\mathrm{s}}$, $V_{\mathrm{c}\mathrm{d}}$, and $V_{\mathrm{c}\mathrm{s}}$, and the Jarlskog invariant $J(V)$. Different cases of specific mass matrices are examined in detail. The quality of the fits for the Fritzsch and Stech type mass matrices is about the same with ${\chi }^2\mathrm{/}\mathrm{d}\mathrm{o}\mathrm{f}=4.23\mathrm{/}3=1.41$ and ${\chi }^2\mathrm{/}\mathrm{d}\mathrm{o}\mathrm{f}=9.10\mathrm{/}4=2.28$, respectively. The fit for a simple generalization (one extra parameter) of the Fritzsch type matrices, in the physical basis, is much better with ${\chi }^2\mathrm{/}\mathrm{d}\mathrm{o}\mathrm{f}=1.89\mathrm{/}4=0.47$. For comparison we also include the results using the quark masses at the 2 GeV energy scale. The fits obtained at this energy scale are similar to that at $M_Z$ energy scale, implying that our results are unaffected by the evolution of the quark masses from 2 to 91 GeV.