  The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias,
Proudfoot and Wakefield, whose properties need to be further explored. In this
paper we prove that the Kazhdan-Lusztig polynomials of fan matroids coincide
with Motzkin polynomials, which was recently conjectured by Gedeon. As a
byproduct, we determine the Kazhdan-Lusztig polynomials of graphic matroids of
squares of paths. We further obtain explicit formulas of the Kazhdan-Lusztig
polynomials of wheel matroids and whirl matroids. We prove the real-rootedness
of the Kazhdan-Lusztig polynomials of these matroids, which provides positive
evidence for a conjecture due to Gedeon, Proudfoot and Young. Based on the
results on the Kazhdan-Lusztig polynomials, we also determine the
$Z$-polynomials of fan matroids, wheel matroids and whirl matroids, and prove
their real-rootedness, which provides further evidence in support of a
conjecture of Proudfoot, Xu, and Young.
