##### Exercise3

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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