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QUESTION

Find the eigenvalues and eigenvectors of $A$ and hence find $A^7$
where

$$ A = \left[\begin{array}{rr}
 29 & -15 \\
 50 & -26 \\
\end{array}\right].
$$

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ANSWER

The eigenvalues are 4 and -1 with eigenvectors
$\left[\begin{array}{c}3\\5\end{array}\right]$ and
$\left[\begin{array}{c}1\\2\end{array}\right]$ respectively.

Hence $A^r=M\Lambda^rM^{-1}$ where $\Lambda=$diag(4,-1), and the
columns of $M$ are the eigenvectors, so

\begin{eqnarray*}
A^r&=&\left[\begin{array}{cc}3&1\\5&2\end{array}\right]
\left[\begin{array}{cc}4^r&0\\0&(-1)^r\end{array}\right]
\left[\begin{array}{cc}2&-1\\-5&3\end{array}\right]\\ &=&
\left[\begin{array}{cc}6\times4^r-5(-1)^r&-3(4^r-(-1)^r)\\
10(4^r-(-1)^r)&-5\times4^r+6(-1)^r\end{array}\right]\\
A^7&=&\left[\begin{array}{cc}98309&-49155\\163850&-81926\end{array}\right]
\end{eqnarray*}






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