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(LinearAlgebra) all 2x2 invertible matrices closed under addition?

  1. Aug 4, 2011 #1
    1. The problem statement, all variables and given/known data

    Suppose V is a vector space.
    Is the set of all 2x2 invertible matrices closed under addition? If so, please prove it. If not, please
    provide a counter-example.

    2. Relevant equations



    3. The attempt at a solution

    well i know that what does it mean to be closed under addition. When V is closed under addition, if I suppose vector u and w are in the V, their addition u+w is also in the V, right?

    The answer for the question is No.
    A counter-example my professor provided is I+(-I)=0
    I and (-I) are invertible, but their addition 0 is not invertible. and I know why it's not invertible.
    But I don't figure out why it is not closed under addition,,.
    If the addition is not invertible, does it mean that the addition is not in the V?
     
  2. jcsd
  3. Aug 4, 2011 #2

    vela

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    Yes.
    Yes. V consists of only invertible matrices, so 0 is not an element in V. So you have u=I and w=-I are both in V, but their sum u+w=0 is not in V. Therefore V is not closed under addition.
     
  4. Aug 4, 2011 #3
    So clear, easy to understand. Thanks!
     
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