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Math Help - should I use real numbers ??

  1. #1
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    should I use real numbers ??

    I have a maths question which reads:

    Let M be the set of all real 2 * 2-matrices of the form:
    (a 0)
    (0 b)

    Show that this set of matrices is closed under matrix multiplication, and that matrix multiplication is commutative on this set.



    Should I use real numbers to solve this problem or use what the question gave??
    Attached Thumbnails Attached Thumbnails should I use real numbers ??-m_multiply.bmp  
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  2. #2
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    Re :should I use real numbers ??

    I forgot to mention.. the attached image is the matrix given in the question.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by kgpretty View Post
    I have a maths question which reads:

    Let M be the set of all real 2 * 2-matrices of the form:
    (a 0)
    (0 b)

    Show that this set of matrices is closed under matrix multiplication, and that matrix multiplication is commutative on this set.



    Should I use real numbers to solve this problem or use what the question gave??
    Are you asking if you should replace a snd b with particular numbers?

    If so the answer is no.

    That M is closed under matrix multiplication is equivalent asking you
    to prove that for any real numbers a, b, c, d, there exist two real numbers
    e,f such that:

    Code:
    [a 0] [c 0] = [e 0]
    [0,b] [0 d]   [0 f]
    and that multiplication is comutative on M equivalent asking you
    to prove that for any real numbers a, b, c, d that:


    Code:
    [a 0] [c 0] = [c 0] [a 0]
    [0,b] [0 d]   [0 d] [0 b]
    RonL
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  4. #4
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    Re: should I use real numbers ??

    Quote Originally Posted by CaptainBlack View Post
    Are you asking if you should replace a snd b with particular numbers?

    If so the answer is no.

    That M is closed under matrix multiplication is equivalent asking you
    to prove that for any real numbers a, b, c, d, there exist two real numbers
    e,f such that:

    Code:
    [a 0] [c 0] = [e 0]
    [0,b] [0 d]   [0 f]
    and that multiplication is comutative on M equivalent asking you
    to prove that for any real numbers a, b, c, d that:


    Code:
    [a 0] [c 0] = [c 0] [a 0]
    [0,b] [0 d]   [0 d] [0 b]
    RonL

    Thankyou for your help.
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  5. #5
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    what exactly does a closed matrix mean? I'm confused.

    I thought it meant.. in this case, that when I did the multiplication.. the resulting matrix would be the same size.
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  6. #6
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    Quote Originally Posted by kgpretty View Post
    what exactly does a closed matrix mean? I'm confused.

    I thought it meant.. in this case, that when I did the multiplication.. the resulting matrix would be the same size.
    Given a set S with an algebraic binary operation *
    Take a subset R of S such that,
    a*b in R for all a,b in R
    Then we say R is closed under the binary operation *

    --
    Informally, closed mean whenever you take two numbers and you operate them you get the same number in the set.

    For example,
    If your matrix ended up being,
    [0 a]
    [b 0]
    For a and b nonzero then your binary operation on this set would not have been closed because you have a diffrenet number (in this case a different matrix).

    This is my 25th Post
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  7. #7
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    I believe I've got it now..

    Here's how I'm going to completely answer the question:

    Code:
    Let a,b,c,d,e,f be real numbers:
    
    [a 0] [c 0] = [e 0]
    [0 b] [0 d] = [0 f]
    
    .: M is closed under matrix multiplication because the product of two M members is also in the M set.
    
    Let A = [a 0]
            [0 a]
    
         B = [b 0]
             [0 b]
    
    AB = [ab   0]
         [0   ab]
    
    BA = [ab   0]
         [0   ab]
    
    AB = BA
    
    .: matrix multiplication is commutative on this set.
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  8. #8
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    I think you meant
    [a 0]
    [0 b]
    Not,
    [a 0]
    [0 b]

    Anyways that is not the biggest problem, you need tos show that,
    [ab 0]
    [0 ba] is an element of these matricies.
    Why?
    Because a,b are real numbers and ab is closed under multiplication. That is the important step you omitted.
    Because you need to show that,
    [ab 0]
    [0 ab]
    is in the set which is true since ab is an element of the set.
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  9. #9
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    Quote Originally Posted by ThePerfectHacker View Post
    I think you meant
    [a 0]
    [0 b]
    Not,
    [a 0]
    [0 b]

    Anyways that is not the biggest problem, you need tos show that,
    [ab 0]
    [0 ba] is an element of these matricies.
    Why?
    Because a,b are real numbers and ab is closed under multiplication. That is the important step you omitted.
    Because you need to show that,
    [ab 0]
    [0 ab]
    is in the set which is true since ab is an element of the set.


    Congrats on your 2500th post!
    How do I show that ab is in M set? Do I just say let a and b be real numbers?
    Last edited by kgpretty; September 17th 2006 at 08:12 AM.
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  10. #10
    Grand Panjandrum
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    To prove commutativity:

    Code:
     
    Let A = [a 0]
            [0 b]
     
         B = [c 0]
             [0 d]
     
    AB = [ac   0]
         [0   bd]
     
    BA = [ca   0]
         [0   db]
    But ordinary multiplication is commutative hence AB=BA,
    and so matrix multiplication is commutative on this set.


    RonL
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