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Math Help - Matrix to the Power 10

  1. #1
    Junior Member
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    Matrix to the Power 10

    I have to calulate A^10.

    I have been given A =
    (1 2
    2 1)

    and B =
    (1 1
    1 -1)

    I have calculated that B^-1AB =
    (3 0
    0 1)

    and I don't really understand what my notes mean to take it on from here. Help would be appreciated.
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  2. #2
    Master Of Puppets
    pickslides's Avatar
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    To find A^{10}

    You must find a diagonal matrix D such that

    A^n= BD^nB^{-1} \implies D^n= B^{-1}A^nB

    In your case n=10

    and where B is the eigenvectors of A

    So first thing to do is find the eigenvalues and eigenvectors of A



    It looks like you have been given that so use

    A^n= BD^nB^{-1}

    As you have D , B and B^{-1}
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  3. #3
    Master Of Puppets
    pickslides's Avatar
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    If that was a little to hard to follow then,

     <br />
\left( {\begin{array}{*{20}{c}}<br />
1& 2 \\<br />
2 & 1\\<br />
\end{array}} \right) ^{10}<br />
= <br />
\left( {\begin{array}{*{20}{c}}<br />
1 & 1 \\<br />
{-1} & 1 \\<br />
\end{array}} \right) <br />
\left( {\begin{array}{*{20}{c}}<br />
3 & 0 \\<br />
0 & 1 \\<br />
\end{array}} \right) ^{10}<br />
\left( {\begin{array}{*{20}{c}}<br />
1 & 1 \\<br />
{-1} & 1 \\<br />
\end{array}} \right) ^{-1}<br />

    Just solve the RHS.
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