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Math Help - Field of diagonal matrices (or not...)

  1. #1
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    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.
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  2. #2
    Senior Member roninpro's Avatar
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    Take R=\mathbb{Z}. Do all of your matrices have multiplicative inverses?
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  3. #3
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    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).
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  4. #4
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    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?
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  5. #5
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    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...
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