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Math Help - 1,1,1,1 Matrix general formula!!!

  1. #1
    Newbie daRitz's Avatar
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    1,1,1,1 Matrix general formula!!!

    please someone help!!

    1) if matrix X=
    |1 1 |
    |1 1 |

    find a general formula for X^n

    2) if matrix Y=
    |1 -1 |
    |-1 1 |

    find a general formula for Y^n

    3) Now find a formula for (X+Y)^n


    4)Let A=aX
    and B=bY

    -find general expressions for A^n, and B^n, using an algebraic method to explain how you arrived at your statement.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi

    Evaluate X^2 and X^3, guess what is X^n and show it using induction.
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  3. #3
    Moo
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    Hello,

    Do it by induction.

    \begin{array}{cccc} X^2 & = & \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} & \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \\ & = & \begin{pmatrix} 2 & 2 \\ 2 & 2 \end{pmatrix} & \end{array}

    Prove by induction that X^n=\begin{pmatrix} ? & ? \\ ? & ? \end{pmatrix}



    Edited
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  4. #4
    Super Member flyingsquirrel's Avatar
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    Quote Originally Posted by Moo View Post
    Hello,
    Prove by induction that X^n=\begin{pmatrix} n & n \\ n & n \end{pmatrix}
    Take a look at X^3, Moo
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  5. #5
    Lord of certain Rings
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    Quote Originally Posted by daRitz View Post
    3) Now find a formula for (X+Y)^n
    Hint: X+Y = I

    For the induction hypothesis in the previous problem, X^n = \begin{pmatrix} 2^{n-1} & 2^{n-1} \\ 2^{n-1} & 2^{n-1} \end{pmatrix}
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  6. #6
    Moo
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    Quote Originally Posted by flyingsquirrel View Post
    Take a look at X^3, Moo


    Oh yeah, 2^{n-1}
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  7. #7
    Newbie daRitz's Avatar
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    does the formula X^n= 2^n-1 work?
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  8. #8
    Moo
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    No, it's X^n=\begin{pmatrix} 2^{n-1} & 2^{n-1} \\ 2^{n-1} & 2^{n-1} \end{pmatrix}
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  9. #9
    Newbie daRitz's Avatar
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    now can anyone help me with all the other problems??
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  10. #10
    Super Member flyingsquirrel's Avatar
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    For Y the idea is exactly the same : evaluate its first powers, guess Y^n and show it by induction.
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  11. #11
    Lord of certain Rings
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    Quote Originally Posted by daRitz View Post
    4)Let A=aX
    and B=bY
    -find general expressions for A^n, and B^n, using an algebraic method to explain how you arrived at your statement.
    A^n = a^n X^n
    You already know X^n, so continue....
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  12. #12
    Newbie daRitz's Avatar
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    you're really helpfull thanks!
    but im alittle confused, am i solving for A^n?

    could you please give me an example by substituting some numbers in?
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  13. #13
    Lord of certain Rings
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    Quote Originally Posted by daRitz View Post
    you're really helpfull thanks!
    but im alittle confused, am i solving for A^n?

    could you please give me an example by substituting some numbers in?
    You know to compute X^n right?(Moo and flyingsquirrel helped you with that)

    So multiply that with a^n.

    You should get

    a^nX^n=a^n \begin{pmatrix} 2^{n-1} & 2^{n-1} \\ 2^{n-1} & 2^{n-1} \end{pmatrix} = \begin{pmatrix} a^n2^{n-1} & a^n2^{n-1} \\ a^n2^{n-1} & a^n2^{n-1} \end{pmatrix}
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  14. #14
    Newbie daRitz's Avatar
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    Talking

    thanks!
    I'm now on the problem "find a general formula for (X+Y)^n

    and like you said, its the identity matrix, and its looking as though it's the same thing as with finding X^n and Y^n, but im not sure how to set up the general formula, any thoughts?
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  15. #15
    Lord of certain Rings
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    Quote Originally Posted by daRitz View Post
    thanks!
    I'm now on the problem "find a general formula for (X+Y)^n

    and like you said, its the identity matrix, and its looking as though it's the same thing as with finding X^n and Y^n, but im not sure how to set up the general formula, any thoughts?
    You know why its called the identity?
    It has a property.... You need not set up a general formula
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