With carpet type mats i usually remove them then beat them,as you might with a rug, before running my vacuum cleaner over them. Car vacs dont tend to be very powerful. Usually does the trick for me.
...mind you, my answer is relevant only to removable mats. When it comes to cleaning the fixed mats, all I do is use a car vacuum and the aforesaid stiff-bristled handbrush.
All good answers. Thanks. But it begs the question why do car manufacturers persist in installing these mats. As far as I can recall, these type of mats have been in all the cars I have had over the years.
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