That's a start -- until now, you haven't told us anything about what you don't understand, so we haven't had any way to help. (Even telling you the whole answer probably wouldn't have given you the understanding you need.)
Maybe you just don't get what the whole problem is asking.
Let's start with the fact that K has to be a group under addition. That means that the sum of any two elements of K must be in K, and that addition must be associative and have an (additive) inverse, right? The latter two are true of any matrices, but you'll have to show that the additive inverse is in the set K.
So what about closure? That means that for any two elements x = ((a, 2b),(b, a)) and y = ((c, 2d),(d, c)), x + y = ((a+c, 2b+2d),(b+d, a+c)) must be able to be written in the same form. Can it? Yes: x + y = ((a+c, 2(b+d)),(b+d, a+c)) = ((m, 2n),(n, m)) where m = a+c and n = b+d. So x + y is in K.
Now do the same sort of thing to show that -x is in K. And so on.
One more thing: does your textbook have any examples of showing that a set is a field? Have you looked through that carefully to see the kinds of thinking that are involved? (Also, I did ask you to state the 7 properties you said you have to prove; please do so, because they might be stated in a different way than I assume. It's important to cooperate with people you ask for help.)