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Math Help - Raising a Matrix to a power, help please!

  1. #1
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    Raising a Matrix to a power, help please!

    So I trying to find a 2 * 2 Matrix M where M^n = I but M^k where k<n does not = I
    So for n = 1 M=I,
    for n = 2 M = \left(\begin{array}{cc}1&-1\\0&-1\end{array}\right)
    for n = 3 M = \left(\begin{array}{cc}1&-1\\3&-2\end{array}\right)
    for n = 4 M = \left(\begin{array}{cc}1&-1\\2&-1\end{array}\right)
    These are just a few examples I found using algebraic methods however I can not find an example for n = 5, is there a possible reason for this?
    Futhermore can you find a 2 * 2 matrix M where M^5 = I and M^{11} = 0 ?
    Also is there a better way to attempting these problems than working through several simultaneous equations?
    Thanks
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  2. #2
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    Quote Originally Posted by Paul616 View Post
    So I trying to find a 2 * 2 Matrix M where M^n = I but M^k where k<n does not = I
    So for n = 1 M=I,
    for n = 2 M = \left(\begin{array}{cc}1&-1\\0&-1\end{array}\right)
    for n = 3 M = \left(\begin{array}{cc}1&-1\\3&-2\end{array}\right)
    for n = 4 M = \left(\begin{array}{cc}1&-1\\2&-1\end{array}\right)
    These are just a few examples I found using algebraic methods however I can not find an example for n = 5, is there a possible reason for this?
    Futhermore can you find a 2 * 2 matrix M where M^5 = I and M^{11} = 0 ?
    Also is there a better way to attempting these problems than working through several simultaneous equations?
    Thanks
    Simultaneous equations probably best.

    Have you looked at 3x3, 4x4, and 5x5 arrays for the similar identities?
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  3. #3
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    Also, regarding the second question -- if you have M^5 = I then M^{11} = M^{10} \cdot M^1 = (M^5)^2 \cdot M = I^2 \cdot M = M \neq 0

    So clearly it isn't possible.
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  4. #4
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    Quote Originally Posted by Paul616 View Post
    So I trying to find a 2 * 2 Matrix M where M^n = I but M^k where k<n does not = I
    So for n = 1 M=I,
    for n = 2 M = \left(\begin{array}{cc}1&-1\\0&-1\end{array}\right)
    for n = 3 M = \left(\begin{array}{cc}1&-1\\3&-2\end{array}\right)
    for n = 4 M = \left(\begin{array}{cc}1&-1\\2&-1\end{array}\right)
    These are just a few examples I found using algebraic methods however I can not find an example for n = 5, is there a possible reason for this?
    for any positive integers m,n you can always find an m \times m matrix M, over \mathbb{C}, such that M^n = I but M^k \neq I for all 0 < k < n. an example is M=\rho I, where \rho is a primitive nth root of unity.
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