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Review problems for exam 1

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  1. Compute the following matrix-vector products.
    1. $\left[\begin{matrix}-3 & 0\\-2 & -4\end{matrix}\right]\left[\begin{matrix}-1\\3\end{matrix}\right]=$
    2. $\left[\begin{matrix}4 & 6\\6 & 0\end{matrix}\right]\left[\begin{matrix}-1\\1\end{matrix}\right]=$

  2. Consider the dynamical system $\vc{x}_{n+1} = A\vc{x}_n$, where $A$ is a $3 \times 3$ matrix. Use the following information about the behavior of the dynamical system to estimate the dominant eigenvalue of $A$ and its eigenvector.

    For the initial condition $\vc{x}_0 = \left[\begin{matrix}1.8\\1.1\\0.8\end{matrix}\right]$, iterating the model forward 10 times gives: \begin{gather*}\vc{x}_1 = \left[\begin{matrix}2.05\\0.38\\2.76\end{matrix}\right], \vc{x}_2 = \left[\begin{matrix}5.216\\0.628\\3.76\end{matrix}\right], \vc{x}_3 = \left[\begin{matrix}7.624\\0.8776\\6.224\end{matrix}\right], \vc{x}_4 = \left[\begin{matrix}12.37\\1.42\\9.704\end{matrix}\right], \vc{x}_5 = \left[\begin{matrix}19.4\\2.225\\15.39\end{matrix}\right]\\ \vc{x}_6 = \left[\begin{matrix}30.7\\3.522\\24.28\end{matrix}\right], \vc{x}_7 = \left[\begin{matrix}48.47\\5.56\\38.36\end{matrix}\right], \vc{x}_8 = \left[\begin{matrix}76.57\\8.783\\60.58\end{matrix}\right], \vc{x}_9 = \left[\begin{matrix}120.9\\13.87\\95.69\end{matrix}\right], \vc{x}_{10} = \left[\begin{matrix}191.0\\21.91\\151.1\end{matrix}\right]\end{gather*}

    For the initial condition $\vc{x}_0 = \left[\begin{matrix}1.8\\1.6\\1.7\end{matrix}\right]$, iterating the model forward 10 times gives: \begin{gather*}\vc{x}_1 = \left[\begin{matrix}3.73\\0.66\\4.17\end{matrix}\right], \vc{x}_2 = \left[\begin{matrix}8.033\\0.966\\6.037\end{matrix}\right], \vc{x}_3 = \left[\begin{matrix}12.16\\1.401\\9.806\end{matrix}\right], \vc{x}_4 = \left[\begin{matrix}19.52\\2.241\\15.37\end{matrix}\right], \vc{x}_5 = \left[\begin{matrix}30.71\\3.522\\24.33\end{matrix}\right]\\ \vc{x}_6 = \left[\begin{matrix}48.56\\5.571\\38.41\end{matrix}\right], \vc{x}_7 = \left[\begin{matrix}76.67\\8.795\\60.67\end{matrix}\right], \vc{x}_8 = \left[\begin{matrix}121.1\\13.89\\95.83\end{matrix}\right], \vc{x}_9 = \left[\begin{matrix}191.3\\21.95\\151.4\end{matrix}\right], \vc{x}_{10} = \left[\begin{matrix}302.2\\34.66\\239.1\end{matrix}\right]\end{gather*}

  3. Find the determinant of the matrix $$\left[\begin{matrix}1 & 0\\2 & 0\end{matrix}\right]$$

  4. Find the eigenvalues of the matrix $$A=\left[\begin{matrix}2 & -6\\2 & 5\end{matrix}\right]$$ Enter the eigenvalues separated by commas (if rounding, use at least 5 significant figures):
    Need help?

  5. Find all solutions to the following system of equations, or explain why none exist. \begin{eqnarray*} - 3 x + 2 y&=&5\\ 9 x - 6 y&=&-15 \end{eqnarray*}

  6. American toad
    American toads (Anaxyrus americanus) start out their lives as tadpoles that live in water, before maturing into adults that live mostly on land.

    Assume that each adult toad produces 40 eggs per year. Adults survive with probability 0.3 each year. Tadpoles survive and mature into adults with probability 0.08, and survive and remain tadpoles with probability 0.07.

    1. Write down a matrix model that describes this population.
    2. Assuming you start with 170 tadpoles and 47 toads, how many tadpoles and toads will you have the following year? (Show your work with matrix vector multiplication.)
    3. Suppose you wanted to know how many tadpoles and toads you had the previous year. Write down the system of equations that you would use to solve this problem and solve this set of equations for the biomasses from the previous year.

      Hint: the matrix equation will look something like $A\vc{x} = \vc{b}$, where $\vc{b} = \begin{bmatrix} 170\\47\end{bmatrix}$.

    4. Find the eigenvalues of your matrix from part (a).
    5. Based on your answer to (c), what can you conclude about the long-term growth of this toad population?

  7. Consider the following $2$-dimensional system of discrete dynamical equations: \begin{eqnarray*} y_{n+1} &=& - 14 y_{n} + 6 z_{n}\\ z_{n+1} &=& - 36 y_{n} + 16 z_{n}\\ y_{0} &=& 6\\ z_{0} &=& -1 \end{eqnarray*}
    1. Compute the next five values of $y$ and $z$.
      $y_1 =$
      ,  $z_1 =$

      $y_2 =$
      ,  $z_2 =$

      $y_3 =$
      ,  $z_3 =$

      $y_4 =$
      ,  $z_4 =$

      $y_5 =$
      ,  $z_5 =$
    2. Convert the system into a matrix equation.

      $\begin{bmatrix} y_{n+1}\\ z_{n+1} \end{bmatrix}=$




      $\begin{bmatrix} y_n\\ z_n\end{bmatrix}$
    3. Find the eigenvalues and eigenvectors of the matrix you found in part a. Enter the eigenvalues separated by commas, in increasing order:

      Enter the eigenvectors in the same order as the corresponding eigenvalues:
      ,
    4. Which eigenvector direction will $(y_n, z_n)$ approach as $n$ goes to infinity?

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Math 2241, Spring 2023