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Math Help - matrix to the power

  1. #1
    Junior Member
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    matrix to the power

    Hi, matrix A=
    -7 18
    -3 8

    I need to find A^2 using A^2=(P)(D^2)(P^-1)
    P=
    2 3
    1 1

    P^-1 =
    -1 3
    1 -2

    D^2 =
    4 0
    0 1

    My problem is how do I multiply out PD^2P^-1 correctly? its doesnt seem to give A^2 as the answer?

    Thanks
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Madrid, Spain
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    Quote Originally Posted by BadMonkey View Post
    Hi, matrix A=
    My problem is how do I multiply out PD^2P^-1 correctly? its doesnt seem to give A^2 as the answer?
    Let us see. Perhaps the origin of the problem is:

    (i) The eigen values of A are \lambda_1=2,\;\lambda_2=-1 (both simple), so A is diagonalizable, being:

    D=\begin{bmatrix}{2}&{0}\\{0}&{-1}\end{bmatrix}

    (ii) In fact, the P matrix satisfies:

    P^{-1}AP=D or equivalently A=PDP^{-1}

    (iii) Then,

    A^2=PDP^{-1}PDP^{-1}=PD^2P^{-1}.

    (iv) So,

    A^2=\begin{bmatrix}{2}&{3}\\{1}&{1}\end{bmatrix}\b  egin{bmatrix}{4}&{0}\\{0}&{1}\end{bmatrix}\begin{b  matrix}{-1}&{3}\\{1}&{-2}\end{bmatrix}=\ldots=\begin{bmatrix}{-5}&{18}\\{-3}&{10}\end{bmatrix}

    Now, find A^2 directly. You'll obtain the same matrix.

    Regards.
    Last edited by FernandoRevilla; November 14th 2010 at 12:00 PM.
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