I currently use all 3 types of media in my various tanks, so can give some insight into which setups they are suitable for.
Purigen is a good alternative to carbon as it only specifically adsorbs excess organic compounds that produce ammonia (which eventually get processed to nitrates). The benefit is it can be used in planted tanks that dose fertilizers without having an impact on the dosed nutrients. Purigen also soaks up tannins too, so its good for making the water crystal clear. It has a limited adsorption lifespan though, so will need to be recharged regularly.
Purigen is good for tanks which have higher bio-loads as it provides additional chemical filtration to adsorb and reduce waste levels.
Matrix and De-nitrate are actually similar products (most people will notice the media look the same but are different size), both are designed to create anaerobic conditions in parts of the media to support denitrifying bacteria... just that De-nitrate is designed to support more of the denitrifying bacteria/process (which converts nitrates into nitrogen gas and release it to the atmosphere), whereas Matrix is designed to support more aerobic bacteria (which convert ammonia to nitrite to nitrates).
De-nitrate requires much lower flow rates to work (ie. less than 200 l/ph), so it is more suitable for very slow flow rate filters, whereas Matrix works better in higher flow rate filters.
Therefore in my perspective, the choice between Matrix and De-nitrate would depend on your tank setup:
If its a tank with lots of fishes, but few (or no) plants and you find the nitrate levels always rising, then using De-nitrate in a slow flow rate filter can help to reduce the nitrate levels (or at least slow down the accumulation of nitrates).
On the otherhand, if your tank has alot of plants with ample filter flow rate and circulation, then it would be better to use Matrix because it specializes in processing organic compounds/waste into nitrates which supply the plants with nutrients (rather than removing the nitrates that plants consume to grow).
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