  In analogy with the classical Kazhdan-Lusztig polynomials for Coxeter groups,
Elias, Proudfoot and Wakefield introduced the concept of Kazhdan-Lusztig
polynomials for matroids. It is known that both the classical Kazhdan-Lusztig
polynomials and the matroid Kazhdan-Lusztig polynomials can be considered as
special cases of the Kazhdan-Lusztig-Stanley polynomials for locally finite
posets. In the framework of Kazhdan-Lusztig-Stanley polynomials, we study the
inverse of Kazhdan-Lusztig-Stanley functions and define the inverse
Kazhdan-Lusztig polynomials for matroids. We also compute these polynomials for
boolean matroids and uniform matroids. As an unexpected application of the
inverse Kazhdan-Lusztig polynomials, we obtain a new formula to compute the
Kazhdan-Lusztig polynomials for uniform matroids. Similar to the
Kazhdan-Lusztig polynomial of a matroid, we conjecture that the coefficients of
its inverse Kazhdan-Lusztig polynomial are nonnegative and log-concave.
