###### Exercise 4

Consider the matrices

\begin{equation*}
A = \left[\begin{array}{cc}
3 \amp 2 \\
-5 \amp -3 \\
\end{array}\right], \qquad
B = \left[\begin{array}{cc}
5 \amp 7 \\
-3 \amp -4 \\
\end{array}\right]\text{.}
\end{equation*}

Find the eigenvalues of \(A\text{.}\) To which of the six types does the system \(\xvec_{k+1}=A\xvec_{k}\) belong?

Using the eigenvalues of \(A\text{,}\) we can write \(A=PEP^{-1}\) for some matrices \(E\) and \(P\text{.}\) What is the matrix \(E\) and what geometric effect does multiplication by \(E\) have on vectors in the plane?

If we remember that \(A^k = PE^kP^{-1}\text{,}\) determine the smallest value of \(k\) for which \(A^k=I\text{?}\)

Find the eigenvalues of \(B\text{.}\)

Then find a matrix \(E\) such that \(B = PEP^{-1}\) for some matrix \(P\text{.}\) What geometric effect does multiplication by \(E\) have on vectors in the plane?

Determine the smallest value of \(k\) for which \(A^k=I\text{.}\)