AP PreCalculus Practice Quiz: Matrices Modeling Contexts

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 15 questions to check your progress.

Question 1 of 15

A city's population is distributed between the city center and the suburbs. Each year, 10% of residents from the city center move to the suburbs, and 5% of residents from the suburbs move to the city center. If the state vector is defined as S = [City Center Population; Suburbs Population], which matrix T correctly models these yearly transitions?

A)[[0.90, 0.05], [0.10, 0.95]]B)[[0.10, 0.05], [0.90, 0.95]]C)[[0.90, 0.10], [0.05, 0.95]]D)[[1.10, 1.05], [0.90, 0.95]]

All Questions (15)

A) [[0.90, 0.05], [0.10, 0.95]]

B) [[0.10, 0.05], [0.90, 0.95]]

C) [[0.90, 0.10], [0.05, 0.95]]

D) [[1.10, 1.05], [0.90, 0.95]]

Correct Answer: A

This question requires constructing a model of a scenario involving transitions between two states. The first column represents people starting in the City Center: 90% stay (0.90) and 10% move to the suburbs (0.10). The second column represents people starting in the Suburbs: 5% move to the City Center (0.05) and 95% stay (0.95). The resulting matrix is [[0.90, 0.05], [0.10, 0.95]].

The transition between two brands of a product, Brand A and Brand B, is modeled by the matrix T = [[0.8, 0.3], [0.2, 0.7]]. The initial state vector, representing the number of customers for each brand, is S_0 = [1000; 2000]. What is the predicted state vector, S_1, after one transition period?

A) [1400; 1600]

B) [800; 1400]

C) [2400; 1800]

D) [1100; 1900]

Correct Answer: A

To predict the future state, we find the product of the transition matrix T and the initial state vector S_0. S_1 = T * S_0 = [[0.8, 0.3], [0.2, 0.7]] * [1000; 2000] = [(0.8*1000 + 0.3*2000); (0.2*1000 + 0.7*2000)] = [800 + 600; 200 + 1400] = [1400; 1600].

In a matrix model describing transitions between states, what does the product of the transition matrix and a corresponding state vector represent?

A) The initial distribution between states.

B) The predicted state after one transition step.

C) The long-term steady state of the system.

D) The state of the system in the past.

Correct Answer: B

As stated in the content, the product of a matrix that models transitions between states and a corresponding state vector can be used to predict future states. Specifically, this product gives the state after one transition.

A student is either 'Prepared' or 'Unprepared' for class each day. The transition is modeled by T = [[0.9, 0.4], [0.1, 0.6]], with the state vector S = [Prepared; Unprepared]. If on Monday the student is Prepared, represented by S_0 = [1; 0], what is the predicted state vector for Wednesday (after two transitions)?

A) [0.81; 0.19]

B) [0.85; 0.15]

C) [0.9; 0.1]

D) [0.78; 0.22]

Correct Answer: B

To predict the state for n=2 transition steps, we can calculate S_2 = T * S_1, where S_1 = T * S_0. First, S_1 = [[0.9, 0.4], [0.1, 0.6]] * [1; 0] = [0.9; 0.1]. Then, S_2 = [[0.9, 0.4], [0.1, 0.6]] * [0.9; 0.1] = [(0.9*0.9 + 0.4*0.1); (0.1*0.9 + 0.6*0.1)] = [0.81 + 0.04; 0.09 + 0.06] = [0.85; 0.15].

The weekly movement of rental cars between two locations, North and South, is modeled by the transition matrix T = [[0.6, 0.2], [0.4, 0.8]]. At the end of this week, there are 400 cars at the North location and 600 at the South location, represented by S_1 = [400; 600]. How many cars were at each location at the start of the week (S_0)?

A) [480; 520]

B) [360; 640]

C) [500; 500]

D) [200; 800]

Correct Answer: C

To predict a past state, we use the product of the inverse of the transition matrix and the current state vector (S_0 = T⁻¹ * S_1). First, find T⁻¹: det(T) = (0.6)(0.8) - (0.2)(0.4) = 0.4. T⁻¹ = (1/0.4) * [[0.8, -0.2], [-0.4, 0.6]] = [[2, -0.5], [-1, 1.5]]. Then, S_0 = [[2, -0.5], [-1, 1.5]] * [400; 600] = [(2*400 - 0.5*600); (-1*400 + 1.5*600)] = [800 - 300; -400 + 900] = [500; 500].

When using a matrix model for state transitions, if repeated multiplication of the transition matrix by successive state vectors results in a state vector that no longer changes with each step, what has the system reached?

A) An inverse state

B) An initial state

C) A transition state

D) A steady state

Correct Answer: D

The content states that repeated multiplication of a matrix that models the transitions between states and corresponding resultant state vectors can predict the steady state. A steady state is a condition where the state vector remains constant from one transition to the next.

A system's transitions are modeled by matrix T. Given the current state vector S_k, which mathematical operation is required to find the state vector from the previous period, S_(k-1)?

A) Multiply T by S_k.

B) Multiply the inverse of T by S_k.

C) Multiply the transpose of T by S_k.

D) Multiply S_k by itself.

Correct Answer: B

According to the provided content, the product of the inverse of a matrix that models transitions between states and a corresponding state vector can predict past states. Therefore, to find S_(k-1), one must calculate T⁻¹ * S_k.

A city's land is zoned as either Residential or Commercial. The transition matrix for zoning changes per decade is T = [[0.9, 0.1], [0.1, 0.9]] for the state vector S = [Residential; Commercial]. The current state is S_1 = [80; 20] in square miles. What does the top number in the vector S_2 = T * S_1 = [74; 26] represent?

A) The number of square miles that changed from Residential to Commercial.

B) The total number of square miles of Residential land after one more decade.

C) The total number of square miles of Commercial land after one more decade.

D) The initial amount of Residential land two decades ago.

Correct Answer: B

The product of the transition matrix and a state vector predicts the next state. Since the state vector is defined as S = [Residential; Commercial], the top number in the resulting vector S_2 represents the predicted amount of Residential land at that future time.

A contextual scenario indicates that the rate of transition from State A to State B is a 15% change per cycle. How would this transition rate be represented in the transition matrix?

A) 15

B) 1.15

C) 0.15

D) -0.15

Correct Answer: C

The content specifies that a contextual scenario can indicate the rate of transitions between states as percent changes. A 15% change is represented as the decimal 0.15 in the matrix.

A scenario is modeled by a transition matrix T and an initial state vector S_0. To predict the state after n transition steps, S_n, which of the following calculations should be performed?

A) n * T * S_0

B) T * (S_0)^n

C) (T⁻¹)^n * S_0

D) T^n * S_0

Correct Answer: D

To apply matrix models to predict future states for n transition steps, the transition matrix T is raised to the power of n and then multiplied by the initial state vector S_0. This is equivalent to repeatedly multiplying by T, n times.

The transition matrix for a system is T = [[0.6, 0.2], [0.4, 0.8]]. After two transitions, the state vector is S_2 = [360; 640]. What was the initial state vector, S_0?

A) [400; 600]

B) [440; 560]

C) [500; 500]

D) [600; 400]

Correct Answer: C

To find a past state, we use the inverse matrix. To go back two steps, we apply the inverse twice: S_0 = T⁻¹ * (T⁻¹ * S_2). First, T⁻¹ = [[2, -0.5], [-1, 1.5]]. Then, S_1 = T⁻¹ * S_2 = [[2, -0.5], [-1, 1.5]] * [360; 640] = [720-320; -360+960] = [400; 600]. Finally, S_0 = T⁻¹ * S_1 = [[2, -0.5], [-1, 1.5]] * [400; 600] = [800-300; -400+900] = [500; 500].

A machine can be in one of two states: 'Working' or 'Broken'. If it is Working, there is a 95% chance it remains Working the next day. If it is Broken, there is a 60% chance it is repaired and is Working the next day. For the state vector S = [Working; Broken], which matrix models this scenario?

A) [[0.95, 0.05], [0.60, 0.40]]

B) [[0.95, 0.40], [0.05, 0.60]]

C) [[0.95, 0.60], [0.05, 0.40]]

D) [[0.05, 0.60], [0.95, 0.40]]

Correct Answer: C

This requires constructing a model. The first column represents the 'Working' state: 95% stay Working (0.95), so 5% become Broken (0.05). The second column represents the 'Broken' state: 60% become Working (0.60), so 40% stay Broken (0.40). The matrix is [[0.95, 0.60], [0.05, 0.40]].

A market is split between two companies, with transitions modeled by T = [[0.7, 0.1], [0.3, 0.9]]. After many transition periods, the market share stabilizes. Which of the following vectors, representing a total of 1000 customers, is the steady-state vector for this system?

A) [500; 500]

B) [700; 300]

C) [100; 900]

D) [250; 750]

Correct Answer: D

A steady-state vector S satisfies the equation T*S = S. We can test the options. For S = [250; 750]: T*S = [[0.7, 0.1], [0.3, 0.9]] * [250; 750] = [(0.7*250 + 0.1*750); (0.3*250 + 0.9*750)] = [175 + 75; 75 + 675] = [250; 750]. Since T*S = S, this is the steady-state vector.

The population of a species is divided into two habitats, Forest and Plains. The transition matrix is T = [[0.8, 0.1], [0.2, 0.9]] for S = [Forest; Plains]. If the current population is S_k = [300; 700], what was the approximate population distribution in the previous period, S_(k-1)?

A) [200; 800]

B) [310; 690]

C) [286; 714]

D) [357; 643]

Correct Answer: C

To find the past state, we calculate S_(k-1) = T⁻¹ * S_k. First, find T⁻¹: det(T) = (0.8)(0.9) - (0.1)(0.2) = 0.72 - 0.02 = 0.7. T⁻¹ = (1/0.7) * [[0.9, -0.1], [-0.2, 0.8]]. Then, S_(k-1) = (1/0.7) * [[0.9, -0.1], [-0.2, 0.8]] * [300; 700] = (1/0.7) * [270 - 70; -60 + 560] = (1/0.7) * [200; 500] = [2000/7; 5000/7] ≈ [286; 714].

A phone's battery is either 'Charging' or 'Not Charging'. The transition matrix is T = [[0.2, 0.9], [0.8, 0.1]] for S = [Charging; Not Charging]. If the phone is currently charging, S_0 = [1; 0], what is the state vector S_1 after one time step?

A) [1; 0]

B) [0.2; 0.8]

C) [0.9; 0.1]

D) [0.8; 0.2]

Correct Answer: B

To predict the next state, we multiply the transition matrix by the current state vector: S_1 = T * S_0. S_1 = [[0.2, 0.9], [0.8, 0.1]] * [1; 0] = [(0.2*1 + 0.9*0); (0.8*1 + 0.1*0)] = [0.2; 0.8]. This represents a 20% chance of remaining in the 'Charging' state and an 80% chance of transitioning to 'Not Charging'.