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Math Help - Matrices

  1. #1
    Super Member Showcase_22's Avatar
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    Matrices

    Let A= \begin{pmatrix}<br />
{4}&{2}\\ <br />
{1}&{1}<br />
\end{pmatrix}. Then what is the value of Det(A^{11}-A^{10})?

    a) -17
    b) -2^9 x 17
    c) -2^{11}
    d) -24
    I have to do this question without a calculator.

    I could work it out doing lots of multiplication but this question is taken off a test where we have to do 11 questions in an hour. There must be a shorter way of doing it.

    Here was another way I was thinking of:

    Det(M- \lambda I)=\begin{vmatrix}<br />
{4-\lambda}&{2}\\ <br />
{1}&{1-\lambda}<br />
\end{vmatrix}=(4-\lambda)(1-\lambda)-2=\lambda^2-5 \lambda+2

    The disciminant is \sqrt{25-8}=\sqrt 17

    I can see that the 17 from the choices is appearing but since it's A^{11} then i'm going to get a \sqrt {17} in my answer somewhere.

    Can anyone help?
    Last edited by Showcase_22; November 12th 2008 at 03:28 AM.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Showcase_22 View Post
    I have to do this question without a calculator.
    But have you tried the question with a calculator?
    Had you done that you would have see some direction.

    For one, the problem as posted has a determinate of 0.
    That cannot be correct. After looking at your work, it is clear that a_{2,1}=1 not 2 as you posted.
    Then the matrix has determinate equal 2.
    Thus your answer is a power of two.
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  3. #3
    Super Member Showcase_22's Avatar
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    Thanks for pointing that out.

    I've changed it now but i'll try it with a calculator.
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