# Thread: Field of diagonal matrices (or not...)

1. ## Field of diagonal matrices (or not...)

Why is the group of nXn diagonal matrices over R not a field with respect to the operations of addition and multiplication of matrices?

Can someone please give an example showing why it is not a field.

Thanks.

2. Take $R=\mathbb{Z}$. Do all of your matrices have multiplicative inverses?

3. Take R = Z. Do all of your matrices have multiplicative inverses?
I don't understand...The question says R not Z. Why would you include complex numbers? And every diagonal matrix has an inverse (except for 0 matrix).

4. 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?

5. O. How silly of me. I just got a bit confused.

Thanks again. You guys rock.

I'm starting calculus next week...uh-oh, I'm gonna be hear a whole lot...