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Math Help - Matrices problem-

  1. #1
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    Matrices problem-

    Consider the matrix M = 3 -1
    4 -1

    Verify that M^2= 2M -I

    the question is what is I ??
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  2. #2
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    Quote Originally Posted by Khonics89 View Post
    Consider the matrix M = 3 -1
    4 -1

    Verify that M^2= 2M -I

    the question is what is I ??
    Generally, when talking about matrices, I is the identity element. ie. the element such that for any matrix A, A*I = A.
    It's the integer equivalent of 1 under multiplication.
    ie. if A is an integer A*1 = A.

    For 2x2 matrices the identity has
    [ 1 0 ]
    [ 0 1 ]

    for 3x3s it's

    [1 0 0 ]
    [0 1 0 ]
    [0 0 1 ]

    And so forth. I don't know how to draw matrices in this thing... sorry...
    Last edited by mr fantastic; May 25th 2009 at 01:48 AM. Reason: Updated quote with new name of OP
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  3. #3
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    Quote Originally Posted by Khonics89 View Post
    Consider the matrix M = 3 -1
    4 -1

    Verify that M^2= 2M -I

    the question is what is I ??
    I assume that you can verify the given result. Then:

    M^2 = 2M - I \Rightarrow M^{-1} M^2 = 2 M^{-1} M - M^{-1} \Rightarrow M = 2I -  M^{-1} \Rightarrow  M^{-1} = 2I - M = \, .... .

    Side note: M^2 = 2M - I \Rightarrow M^2 - 2M + I = 0 has been got using the Cayley-Theorem Theorem: Cayley-Hamilton Theorem


    Edit: Whoops, I just realised that your question wasn't find the inverse of M. Not to worry, since you asked what I is I figure I've probably anticipated what your next question is.
    Last edited by mr fantastic; May 25th 2009 at 01:48 AM.
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  4. #4
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    Surely that's more complicated than necessary.

    If M^2= 2M- I then M^2- 2M= M(M- 2I)= I so M^{-1}= M-2I by the definition of "inverse matrix"- its product with M is the identity matrix.
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  5. #5
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    First square M and you get:

    5 -2
    8 -3

    Then multiply M by 2 you get:

    6 -2
    8 -2

    Finally take away the identity matrix I away from 2M:

    |6 -2| - |1 0|
    |8 -2| |0 1|

    Whcih gives you the same valve as M^2..........
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  6. #6
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    Quote Originally Posted by HallsofIvy View Post
    Surely that's more complicated than necessary.

    If M^2= 2M- I then M^2- 2M= M(M- 2I)= I Mr F says: *Ahem* .... = -I, surely ....
    so M^{-1}= M-2I by the definition of "inverse matrix"- its product with M is the identity matrix.
    Well, one point in its favor is that the more complicated way appears less prone to error .....
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  7. #7
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    Don't you have a "smilie" for "abject surrender"? It would fit here.

    (I considered "mooning" for a second but thought better of it.)
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