# Sum of two unitary matrix is equal to every matrix? [closed]

Let $R=M_{n}(Z_{2})$, can we write every matrices of $R$ as sum of two matrices of $GL_{n}(Z_{2})$?

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## closed as not a real question by Victor Protsak, Loop Space, Akhil Mathew, Yemon Choi, Qiaochu YuanAug 23 '10 at 21:20

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

The body of the question is unclear and doesn't correspond to the title. –  Victor Protsak Aug 23 '10 at 19:26
However, if a question like this appears unclear, I would rather ask the questioner to clarify it. In the present case, the only unclear point seems to me the use of "unitary" in the title, in place of "invertible". –  Pietro Majer Aug 24 '10 at 19:13

It appears you are asking whether $R$ is a 2-good ring. The answer is yes. You may find the paper "2-good rings" by Peter Vamos to be useful in giving you some background information.
I failed to remark that I am, of course, assuming that $n>1$, otherwise the result is not true. I also want to mention that a colleague of mine Thomas Dorsey (along with two coauthors) has a preprint on this topic which is quite good. You might look for that paper to appear soon. –  Pace Nielsen Aug 24 '10 at 17:28