Matrix Powers, Dynamical Systems and Steady States Worksheet with Answer Key

What's on the worksheet

1. For each matrix A, find the eigenvalues and describe the origin for x_k+1 = Ax_k:
   \begin{bmatrix} 0.4 & 0.3 \\ 0.1 & 0.6 \end{bmatrix}\begin{bmatrix} 0.1 & 0.7 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} 1 & 2 \\ 3 & 0 \end{bmatrix}\begin{bmatrix} 1 & -1 \\ 1 & 0.5 \end{bmatrix}
2. Consider the system x_k+1 = Ax_k with A = \begin{bmatrix} -7 & 2 \\ -10 & 2 \end{bmatrix}, \qquad \mathbf{v}_1 = \begin{bmatrix} 1 \\ 2…
3. x y \lambda = \tfrac{3}{2} \lambda = \tfrac{1}{2} P Q R S A 2 × 2 matrix A has eigenvector v_1 = (2, 1) for _1 = 32 and eigenvector v_2 = (1, -1)…
4. A hospital keeps 600 wheelchairs for visitors at its two entrances, the North entrance and the South entrance.
5. A creamery sells milk in 950 returnable glass bottles through three shops:
6. A marine reserve models a made-up seahorse population in two stages, juveniles j_k and adults a_k in year k.
7. Let A be an n × n column-stochastic matrix.