# Math Insight

### Quiz 2

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Total points: 1
1. Find the eigenvalues and eigenvectors of the matrix $$A=\left[\begin{matrix}-1 & -2\\1 & 2\end{matrix}\right]$$ Enter the eigenvalues separated by commas, in increasing order:

Enter the eigenvectors in the same order as the corresponding eigenvalues:
,

2. Let $r_1$, $r_2$ and $r_3$ represent the activity of three neurons in the network illustrated to the right. (We use $r$ because these state variables represent the rate at which the neurons fire their output spikes.) We record the activity of the neurons every 100 ms, and let $t$ index these 100 ms time bins so that $t=0$ represents the activity at time 0 ms, $t=1$ represents the activity at time 100 ms, etc. We model the dynamics of the network by the following discrete dynamical system. $$\begin{bmatrix}r_1\\r_2\\r_3\end{bmatrix}_{t+1}= \left[\begin{matrix}0.3 & 0.15 & 0\\0 & 0.3 & 0.03\\0.12 & 0.13 & 0.3\end{matrix}\right]\begin{bmatrix}r_1\\r_2\\r_3\end{bmatrix}_{t}, \qquad \text{for t=0,1,2, \ldots}$$
1. The number $0.3$ appears in the matrix three times. What is the biological meaning of that number? It indicates how the firing rate of a neuron at time $t$ influences the firing rate of
at time
.

Interpret the biological meaning of the entry $0.03$ in the above matrix.
It indicates how the firing rate of neuron
at time $t$ influences the firing rate of neuron
at time
.

2. After a long time, what will be the ratio of the firing rate of neuron $2$ to the firing rate of neuron $1$?

Include at least 5 significant digits in your response.

3. The emerald ash borer (Agrilus planipennis) is a beetle that is invasive to North America, causing the destruction of ash trees. You decide to build a matrix model to describe the emerald ash borer population. Suppose each juvenile beetle (n1) survives and matures into an adult the following year with probability $0.09$, and survives but remains a juvenile with probability $0.31$. Each adult beetle (n2) survives each year with probability $0.23$. Each adult produces $49$ new juvenile beetle offspring each year.
1. Fill in the matrix model below, based on this data.

$\displaystyle\begin{bmatrix} n1 \\ n2 \end{bmatrix}_{t+1}=$
$\displaystyle\begin{bmatrix} n1\\ n2 \end{bmatrix}_t$
2. Calculate the eigenvalues for this model.
$\lambda_1 =$
, $\lambda_2 =$

(Enter in any order. Include at least 5 significant digits.)
3. What can you conclude about the population size of emerald ash borers over time?
Eventually, by what factor will the population size increase or decrease each year?

4. Find the determinant of the matrix $$\left[\begin{matrix}5 & 4\\-4 & 4\end{matrix}\right]$$

5. Use the following applet to find the eigenvectors of \begin{align*} A=\left[\begin{matrix}-0.309 & 0.873\\-0.364 & -1.691\end{matrix}\right]. \end{align*} The dashed vectors represent the vectors that result from multiplying $A$ by the solid vectors.
Feedback from applet
eigenvectors:
Matrix-vector multiplication (Show)

Enter the eigenvectors:
,

Enter the eigenvalues corresponding to the eigenvectors (be sure to enter them in the same order as the eigenvectors and include at least two significant figures). You can use the matrix-vector multiplication section to determine the result of multiplying $A$ by the two vectors in the applet.
Corresponding eigenvalues:
,