We consider new kinds of max and min matrices, $\left[ a_{\max(i,j)}\right] _{i,j\geq1}$ and $\left[ a_{\min(i,j)}\right] _{i,j\geq1},$ as generalizations of the classical max and min matrices. Moreover, their reciprocal analogues for a given sequence $\left\{ a_{n}\right\} $ have been studied. We derive their $LU$ and Cholesky decompositions and their inverse matrices as well as the $LU$-decompositions of their inverses. Some interesting corollaries will be presented.