When building a box, you want its resonant frequency to be outside the frequency window the subwoofer system will be expected to play. In other words, if the sub is gonna play 20-100 hz, you dont wan the box's natural resonant freq to be 60hz.
Knowing this, we must now look at how to achieve our goal. The box must be built to either resonate above 100hz, or below 20hz (for my example box). You lower res freq by adding mass, which is what the OP is suggesting by adding mat to the inside of the box.
The second way is to raise the res freq of the box. You do this by making it rigid. You can have a light, but strong box, that vibrates at a relatively high frequency (outside your playing range). You increase enclosure rigidity by, you guessed it, adding bracing.
Some home audio systems are designed to go below the operating range of the sub, by building the enclosure out of concrete. That works in a livingroom or basement, but not so much in a trunk. So the vast majority of car audio boxes are designed to be rigid, to raise res freq above the operating range... many times with the designer not even realizing it.
So knowing all this, you can now see why adding a mass loading mat to the inside of your box will be nothing but a waste of materials. The foams mentioned above are meant strictly for dispersing back waves, do not confuse that with deadening.