  We present a formula which relates the Kazhdan-Lusztig polynomial of a
matroid $M$, as defined by Elias, Proudfoot and Wakefield, to the
Kazhdan--Lusztig polynomials of the matroid obtained by deleting an element,
and various contractions and localizations of $M$. We give a number of
applications of our formula to Kazhdan--Lusztig polynomials of graphic
matroids, including a simple formula for the Kazhdan--Lusztig polynomial of a
parallel connection graph.
