Q: Which of the following statements about matrix multiplication is CORRECT?

You can multiply a $$3 \times 2$$ matrix by a $$2 \times 3$$ matrix, and the result is a $$3 \times 3$$ matrix.

Q: Given $$A = \left[ \begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array} \right]$$ and $$B = \left[ \begin{array}{cc} 0 & 1 \\ 0 & 1 \end{array} \right]$$, what is the result of $$A + B$$?

$$\left[ \begin{array}{cc} 1 & 3 \\ 3 & 5 \end{array} \right]$$

Question 1

Given the discrete dynamical system (DDS) $$P_{n+1} = 0.8P_n + 20$$, which of the following statements is TRUE about its equilibrium and stability?

Accepted Answer

The equilibrium is $$P_e = 100$$, and it is stable.

Question 2

Which of the following sequences is an explicit sequence?

Accepted Answer

$$x_n = 5(1.2)^n$$

Question 3

A population of bacteria is modeled by the Malthusian growth model with immigration: $$P_n = (1 + r)P_{n-1} + \mu. If$$ $$r = 0.2$$ and $$\mu = 50$$, what is the explicit formula for $$P_n in $$terms of $$P_0$$?

Accepted Answer

$$P_n = 1.2^n (P_0$ - 250) + 250$$

Question 4

Consider the logistic model $$p_n = p_{n-1} + r p_{n-1} (1 - \frac{p_{n-1}}{N}). If$$ $$r = 0.5$$ and $$N = 1000$$, which of the following is TRUE about its equilibria?

Accepted Answer

There are two equilibria: $$p_e = 0 ($$unstable) and $$p_e = 1000 ($$stable).

Question 5

Given the system of equations represented by the augmented matrix $$\left[ \begin{array}{ccc|c} 1 & 0 & 13 & 39 \ 0 & 1 & 3 & -15 \ 0 & 0 & 0 & 0 \end{array} ight]$$, what is the general solution in terms of the free variable $$z$$?

Accepted Answer

$$(x, y, z) = (39 - 13z, -15 - 3z, z)$$

Question 6

Which of the following statements about matrix multiplication is CORRECT?

Accepted Answer

You can multiply a $$3 	imes 2$$ matrix by a $$2 	imes 3$$ matrix, and the result is a $$3 	imes 3$$ matrix.

Question 7

Suppose you have the DDS $$P_{n+1} = -1.5P_n + 25$$ with $$P_0 = 8$. What is the explicit formula for P_n$?

Accepted Answer

$$P_n = (-1.5)^n (8 - 10) + 10$$

Question 8

A farmer has 21,800 lbs of nitrogen, 29,600 lbs of phosphate, 550 hours of labor, and 160 acres for planting apples ($$a$$), broccoli ($$b$$), and carrots ($$c). $$Which equation correctly represents the nitrogen constraint?

Accepted Answer

$$120a + 150b + 180c = 21,800$$

Question 9

Given $$A = \left[ \begin{array}{cc} 1 & 2 \ 3 & 4 \end{array} ight]$$ and $$B = \left[ \begin{array}{cc} 0 & 1 \ 0 & 1 \end{array} ight]$$, what is the result of $$A + B$$?

Accepted Answer

$$\left[ \begin{array}{cc} 1 & 3 \ 3 & 5 \end{array} ight]$$

Question 10

Which of the following is a necessary condition for two matrices $$A$$ and $$B to be $$multiplied as $$AB$$?

Accepted Answer

The number of columns of $$A$$ must equal the number of rows of $$B.$$

Question 11

Consider the DDS $$P_{n+1} = P_n^2$. Which of the following is TRUE?

Accepted Answer

It is a non-linear DDS with equilibrium at $$P_e = 0$$ and $$P_e = 1.$$

Question 12

Given the logistic model $$p_n = 2.7p_{n-1} - 0.0001p_{n-1}^2$$, what is the carrying capacity $$N$$?

Accepted Answer

$$N = 17,000$$

Question 13

Which of the following augmented matrices represents a system with no solution?

Accepted Answer

$$\left[ \begin{array}{ccc|c} 0 & 0 & 0 & 5 \end{array} \right]$$

Question 14

If a $$3 	imes 2$$ matrix is multiplied by a $$2 	imes 3$$ matrix, what is the dimension of the resulting matrix?

Accepted Answer

$$3 	imes 3$$

Question 15

Which of the following is NOT an elementary row operation in the Gauss-Jordan method?

Accepted Answer

Swapping two columns.

Question 16

Given the linear DDS $$x_n = ax_{n-1} + b$$ with $$|a| \u003c 1$$, which statement is TRUE?

Accepted Answer

The equilibrium $$x_e is $$stable.

Question 17

Which of the following systems of equations corresponds to the augmented matrix $$\left[ \begin{array}{ccc|c} 3 & 0 & 2 & 37 \ 3 & 1 & 0 & 40 \ 5 & 0 & -3 & 19 \end{array} ight]$$?

Accepted Answer

$$\begin{cases} 3x + 0y + 2z = 37 \ 3x + y + 0z = 40 \ 5x + 0y - 3z = 19 \end{cases}$$

Study Guide: Discrete Dynamical Systems, Systems of Equations, and Matrices

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