<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}

\begin{document}

\parindent=0pt


QUESTION

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

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

\bigskip

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






\end{document}
</pre></body></html>