I agree with Geoduck (because I actually understand what he is saying, wheras the rest of you evidently don't).
Yes, of course Мать and Мат sound different and distinct, no-one has said otherwise, but the fact is that any learner [looks at Tatu] who spends more than five seconds worrying about that difference (or the difference between и and ы, or щ and ш, or any other of the reccuring themes in the Pronunciation forum) in the initial stages of study are simply wasting their own time. It's totally pointless.
The reason is, you categorically aren't going to sound like anything other than a foreigner and a beginner until you have spoken and listened to natives for a considerable amount of time anyway. For your first unpteen thousand conversations you are going to be a stammering idiot who can't get his declensions right, muddles up his conjugations, constantly forgets words, misunderstands questions directed at himself, etc etc. How accurate your accent is simply isn't going to be an issue. You are much better to get your pronunciation sort-of-right, then spend your finite time doing something that will actually make a difference to how comprehesible you are, such as reading, practicing sentence formation, conversing with a fluent speaker, listening to Russian radio or watching Russian films, and so on.
Russians aren't stupid (no more stupid than anyone else anyway), and no Russian is ever going to hold a missplaced hard consononant against you, or stare blankly for more than two seconds should you mistakenly ask for manhole soup instead of onion.
And by the time you do reach a stage where your grasp of grammar and vocabularly are such that you can carry a normal conversation without making regular mistakes or pauses for translation, by then the difference between hard and soft consonants will be as natural to you as it is to natives.
That's if you ever reach that stage.