I don't think it would hold up for very many washes. You could, however, wait until the end of a wash cycle when the machine spins all the water out and then trap the unrolled mat against the side of the washing machine with clothes still in there and then let the machine spin-cycle out. This would force the wash liquid with soap and whatnot through the mat without subjecting it to a full agitation cycle. Or just throw it in there and it would probably be fine for several washes lol
When I need to wash my mats, I just use a scrub pad, and some lightly soapy water. Scrub the whole surface, then go back over with a clean scrub pad. Pat it dry to get most of the liquid out, then let air dry. It'll be nice and clean after.