Exercise4

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

  2. 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?

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

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

  5. 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?

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

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