Take . Do all of your matrices have multiplicative inverses?
roninpro misunderstood. He thought that by saying that "over R not a field" you were trying to take a the matrics "over" something other than a field. Of course, you meant you want to show that the matrices of R, the field of real numbers, is not itself a field.
To answer your question, look at any diagonal matrix that has some zero entries and some non-zero entries on the field. Since it has non-zero entries, it is not the 0 matrix and, if this were field would have to have and inverse. Why does such a matrix not have an inverse?