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Math 2890 Sample Exam Problems

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Exam III

3.3      Cramer’s Rule, Volume, and Linear Transformations
 
4.8      Applications to Difference Equations
4.9      Applications to Markov Chains

5.1      Eigenvectors and Eigenvalues
5.2      The Characteristic Equation
5.3      Diagonalization
5.5      Complex Eigenvalues
         Schur Decomposition
5.6      Discrete Dynamical Systems
5.7      Applications to Differential Equations
5.8      Iterative Estimates for Eigenvalues
  1. Apply a nonrecursive filter to a signal.
  2. Find the general solution of a difference equation.
  3. Find the steady state vector of a stochastic matrix.
  4. Find `A^k` given `A=X Lambda X^{-1}`.
  5. Find an eigenvector for a matrix given an eigenvalue.
  6. Find the eigenvalue for a matrix given an eigenvector.
  7. Find the eigenvalues of a matrix.
  8. Create a matrix with prescribed eigenvalues and eigenvectors.
  9. Create a matrix with prescribed properties.
  10. Diagonalize a matrix.
  11. Use the Power Method to estimate the dominant eigenvalue and its eigenvector for a matrix.
  12. Use the Inverse Power Method to estimate the smallest eigenvalue and its eigenvector for a matrix.
  13. Analyze the behavior of the discrete dynamical system `x_{k+1}=A x_k.`
  14. Analyze the behavior of the differential equation `y'=A y.`
  15. Solve a Discrete Dynamical System.
  16. Solve an Initial Value Problem.
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