The Clipped Courtyard

A courtyard is paved with $64$ square tiles in an $8 \times 8$ grid, colored alternately slate and limestone like a checkerboard. A landscaper has rectangular mats, each exactly the size of two adjacent tiles $(2 \times 1)$, and $32$ of them would carpet the courtyard perfectly. Renovations then remove the two corner tiles at diagonally opposite ends of the courtyard. Can the remaining $62$ tiles be carpeted exactly by $31$ mats?