Okay, here's the "trashing": Your "all 0" matrices make no sense. The "0 matrix" can't be a basis vector since any set of matrices containing the 0 matrix can't be independent.

a+ d= b+ c gives one condition on the four numbers. You can solve for one of the numbers in terms of the other three. For example, you can write d= b+ c- a. I have no idea what you mean by "e= epsilon". What is epsilon?

Here's how I would do the problem:

Taking a= 1, b= c= 0, d= -1 so is in the space.

Taking b= 1, a= c= 0, d= 1 so is in the space.

Taking c= 1, a= b= 0, d= 1 so is in the space.

The space has dimension 3 and those three matrices form a basis.

Of course, you could also solve for b, c, or d and get a different, but equally correct, basis.