Q: Given the system of equations: $$\begin{cases} 2x + 3y = 5 \\ 4x - y = 6 \end{cases}$$ Which matrix equation represents this system?

$$\begin{pmatrix} 2 & 3 \\ 4 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 6 \end{pmatrix}$$

Q: A population of rabbits is modeled by the Leslie matrix: $$\begin{pmatrix} 0 & 2 \\ 0.5 & 0 \end{pmatrix} If $$the initial state vector is $$\begin{pmatrix} 100 \\ 50 \end{pmatrix}$$, what is the population vector after one time step?

$$\begin{pmatrix} 0 \times 100 + 2 \times 50 \\ 0.5 \times 100 + 0 \times 50 \end{pmatrix} = \begin{pmatrix} 100 \end{pmatrix} \begin{pmatrix} 50 \end{pmatrix}$$

Q: Which of the following is the determinant of the matrix $$\begin{pmatrix} 4 & 7 \\ 2 & 6 \end{pmatrix}$$?

$$4 \times 6 - 7 \times 2 = 10$$

Q: Which of the following vectors is an eigenvector of the matrix $$\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$$?

$$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$$

Q: A Markov matrix is used to model the distribution of customers between two stores. If the initial state vector is $$\begin{pmatrix} 0.6 \\ 0.4 \end{pmatrix}$$ and the Markov matrix is $$\begin{pmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{pmatrix}$$, what is the distribution after one time step?

$$\begin{pmatrix} 0.7 \times 0.6 + 0.3 \times 0.4 \\ 0.3 \times 0.6 + 0.7 \times 0.4 \end{pmatrix} = \begin{pmatrix} 0.54 \\ 0.46 \end{pmatrix}$$

Q: If a matrix $$A is$$ $$2 \times 3$$ and matrix $$B is$$ $$3 \times 2$$, which of the following is true about their multiplication?

The product $$AB is a$$ $$2 \times 2$$ matrix.

Question 1

Which of the following best describes a Markov matrix?

Accepted Answer

A matrix where each row sums to 1 and all entries are non-negative.

Question 2

Given the system of equations:

$$\begin{cases} 2x + 3y = 5 \ 4x - y = 6 \end{cases}$$

Which matrix equation represents this system?

Accepted Answer

$$\begin{pmatrix} 2 & 3 \ 4 & -1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 5 \ 6 \end{pmatrix}$$

Question 3

A population of rabbits is modeled by the Leslie matrix:

$$\begin{pmatrix} 0 & 2 \ 0.5 & 0 \end{pmatrix}

If $$the initial state vector is $$\begin{pmatrix} 100 \ 50 \end{pmatrix}$$, what is the population vector after one time step?

Accepted Answer

$$\begin{pmatrix} 0 	imes 100 + 2 	imes 50 \ 0.5 	imes 100 + 0 	imes 50 \end{pmatrix} = \begin{pmatrix} 100 \end{pmatrix} \begin{pmatrix} 50 \end{pmatrix}$$

Question 4

Which of the following is the determinant of the matrix $$\begin{pmatrix} 4 & 7 \ 2 & 6 \end{pmatrix}$$?

Accepted Answer

$$4 	imes 6 - 7 	imes 2 = 10$$

Question 5

Which of the following vectors is an eigenvector of the matrix $$\begin{pmatrix} 2 & 0 \ 0 & 3 \end{pmatrix}$$?

Accepted Answer

$$\begin{pmatrix} 1 \ 0 \end{pmatrix}$$

Question 6

A Markov matrix is used to model the distribution of customers between two stores. If the initial state vector is $$\begin{pmatrix} 0.6 \ 0.4 \end{pmatrix}$$ and the Markov matrix is $$\begin{pmatrix} 0.7 & 0.3 \ 0.3 & 0.7 \end{pmatrix}$$, what is the distribution after one time step?

Accepted Answer

$$\begin{pmatrix} 0.7 	imes 0.6 + 0.3 	imes 0.4 \ 0.3 	imes 0.6 + 0.7 	imes 0.4 \end{pmatrix} = \begin{pmatrix} 0.54 \ 0.46 \end{pmatrix}$$

Question 7

Which of the following is the characteristic polynomial of the matrix $$\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$$?

Accepted Answer

$$\$$lambda^2$$ - 5\lambda + 2$$

Question 8

If a matrix $$A is$$ $$2 	imes 3$$ and matrix $$B is$$ $$3 	imes 2$$, which of the following is true about their multiplication?

Accepted Answer

The product $$AB is a$$ $$2 	imes 2$$ matrix.

Question 9

A normalized eigenvector is:

Accepted Answer

An eigenvector whose length (magnitude) is 1.

Question 10

In the context of population modeling, what does the principal eigenvalue of a Leslie matrix represent?

Accepted Answer

The long-term growth rate of the population.

Question 11

Which of the following is NOT a property of the identity matrix?

Accepted Answer

The identity matrix has all entries equal to 1.

Question 12

If $$A = \begin{pmatrix} 2 & 1 \ 0 & 3 \end{pmatrix}$$ and $$\vec{v} = \begin{pmatrix} 1 \ 0 \end{pmatrix}$$, what is $$A \vec{v}$$?

Accepted Answer

$$\begin{pmatrix} 2 \ 0 \end{pmatrix}$$

Question 13

A chemical reaction is modeled by a Markov matrix. If the steady state vector is $$\begin{pmatrix} 0.2 \ 0.8 \end{pmatrix}$$, what does this represent?

Accepted Answer

The proportion of molecules in each state after a long time.

Question 14

Which of the following is the inverse of the matrix $$\begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}$$?

Accepted Answer

$$\begin{pmatrix} -2 & 1 \ 1.5 & -0.5 \end{pmatrix}$$

Question 15

If a matrix $$A$$ has a dominant eigenvalue $$\lambda = 1.2$$, what does this imply about the long-term behavior of a population modeled by $$A$$?

Accepted Answer

The population will grow over time.

Question 16

Which of the following is a correct way to multiply a matrix by a scalar?

Accepted Answer

Multiply each entry of the matrix by the scalar.

Question 17

A hospital tracks patient movement between two wards using a Markov matrix. If the matrix is $$\begin{pmatrix} 0.8 & 0.2 \ 0.1 & 0.9 \end{pmatrix}$$, what is the probability that a patient in ward 2 remains in ward 2 after one time step?

Accepted Answer

0.9

Question 18

Which of the following is the characteristic equation for the matrix $$\begin{pmatrix} 5 & 2 \ 1 & 3 \end{pmatrix}$$?

Accepted Answer

$$\$$lambda^2$$ - 8\lambda + 13 = 0$$

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