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Math Help - subsets of M3R

  1. #1
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    subsets of M3R

    Let G be a subset of Msub3(R) satisfying the following three conditions:

    1. For every A and B in G, the matrix product AB is in G
    2. There is a matrix E in G such that EA = A = AE for all A in G
    3. For every A in G, there exists a B in G such that AB = E = BA

    I'm supposed to figure out if the following statements are true or false. Prove our counterexample them.

    1. E is the 3 x 3 identity matrix.
    2. If A is in G then Det(A) is not equal to zero.

    My feeling is that 1 is true, but I don't know how to prove it. And I'm not sure about 2.
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  2. #2
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    These are properties of a group

    For Q1, try multiplying your matrix E by the identity.

    For Q2, take the determinant of the product of matrices in property 3. Suppose the determinant of A were 0. Is this possible?
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  3. #3
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    Quote Originally Posted by grandunification View Post
    Let G be a subset of Msub3(R) satisfying the following three conditions:

    1. For every A and B in G, the matrix product AB is in G
    2. There is a matrix E in G such that EA = A = AE for all A in G
    3. For every A in G, there exists a B in G such that AB = E = BA

    I'm supposed to figure out if the following statements are true or false. Prove our counterexample them.

    1. E is the 3 x 3 identity matrix.
    2. If A is in G then Det(A) is not equal to zero.

    My feeling is that 1 is true, but I don't know how to prove it. And I'm not sure about 2.

    The set G=\{0\} \in\,M_3(\mathbb{R}) fulfills the conditions and (1)-(2) are false.

    Tonio
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  4. #4
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    Ya, that's what I was gonna due, but I doubted myself.

    There's still a slight problem though: namely that you want me to take EA = A = AE for all A in G, and use I for A. But how do we know that I is in G?

    I guess you could say:
    1. Let B be in G then IB = B which is in G by assumption.
    2. EI = I = IE for E = I.
    3. (I)(I) = I = (I)(I)
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