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AP Course

AP Precalculus

Study nine focused AP Precalculus units through topic lessons, original graph examples, quizzes, and practice sets.

Follow the College Board topic sequence across polynomial, rational, exponential, logarithmic, trigonometric, polar, parametric, vector, and matrix functions.

Lessons
Change in TandemRates of ChangeRates of Change in Linear and Quadratic FunctionsPolynomial Functions and Rates of ChangePolynomial Functions and Complex ZerosPolynomial Functions and End BehaviorRational Functions and End BehaviorRational Functions and ZerosRational Functions and Vertical AsymptotesRational Functions and HolesEquivalent Representations of Polynomial and Rational ExpressionsTransformations of FunctionsFunction Model Selection and Assumption ArticulationFunction Model Construction and ApplicationChange in Arithmetic and Geometric SequencesChange in Linear and Exponential FunctionsExponential FunctionsExponential Function ManipulationExponential Function Context and Data ModelingCompeting Function Model ValidationComposition of FunctionsInverse FunctionsLogarithmic ExpressionsInverses of Exponential FunctionsLogarithmic FunctionsLogarithmic Function ManipulationExponential and Logarithmic Equations and InequalitiesLogarithmic Function Context and Data ModelingSemi-log PlotsPeriodic PhenomenaSine, Cosine, and TangentSine and Cosine Function ValuesSine and Cosine Function GraphsSinusoidal FunctionsSinusoidal Function TransformationsSinusoidal Function Context and Data ModelingThe Tangent FunctionInverse Trigonometric FunctionsTrigonometric Equations and InequalitiesThe Secant, Cosecant, and Cotangent FunctionsEquivalent Representations of Trigonometric FunctionsTrigonometry and Polar CoordinatesPolar Function GraphsRates of Change in Polar FunctionsParametric FunctionsParametric Functions Modeling Planar MotionParametric Functions and Rates of ChangeParametrically Defined Circles and LinesImplicitly Defined FunctionsConic SectionsParametrization of Implicitly Defined FunctionsVectorsVector-Valued FunctionsMatricesThe Inverse and Determinant of a MatrixLinear Transformations and MatricesMatrices as FunctionsMatrices Modeling Contexts
Quizzes
Practice Problems Open a unit to choose from 58 topic-aligned practice sets.

Unit 9 · Topic 4.14 · Optional

Matrices Modeling Contexts

Model repeated movement between two states with a transition matrix, predict future distributions, identify a steady state, and use an inverse matrix to reconstruct a possible past state.

Learning Goals

  • Translate state-retention and switching percentages into a transition matrix.
  • Use a consistent row or column convention and interpret every matrix entry.
  • Predict future states with repeated matrix multiplication and matrix powers.
  • Check totals, units, nonnegativity, and contextual reasonableness.
  • Identify and verify a steady-state distribution.
  • Use an inverse transition matrix to estimate a previous state when possible.

1. Discrete State Models

A state model tracks how a population or quantity is divided among categories at equally spaced steps. For two states A and B, use the column state vector

\[\mathbf p_n=\begin{bmatrix}A_n\\B_n\end{bmatrix},\]

where n might count days, months, semesters, or another discrete interval. The entries may be counts, proportions, or percentages, but their units must remain consistent.

2. Read Percentages as Movements

StatementTransition meaning
80% of A remains in AA to A: 0.80
20% of A switches to BA to B: 0.20
30% of B switches to AB to A: 0.30
70% of B remains in BB to B: 0.70

For each source state, the outgoing percentages should total 100% when no members enter or leave the system.

3. A Transition Diagram

Matrices Modeling Contexts example graphThe transition matrix redistributes the two-state population each step.
The transition matrix redistributes the two-state population each step.

Loops represent the fraction that remains in the same state. Arrows between the boxes represent the fraction that switches states during one step.

4. Build the Transition Matrix

Using column state vectors, let each column describe where one source state goes:

\[M=\begin{bmatrix} 0.80&0.30\\ 0.20&0.70 \end{bmatrix}.\]
From AFrom B
To A0.800.30
To B0.200.70

Each column sums to 1. A row-vector convention is also possible, but then the matrix layout and multiplication order must change; never mix conventions.

5. Predict the Next State

If initially all 100 members are in A,

\[\mathbf p_0=\begin{bmatrix}100\\0\end{bmatrix}.\]

After one step,

\[\mathbf p_1=M\mathbf p_0 =\begin{bmatrix}0.8&0.3\\0.2&0.7\end{bmatrix} \begin{bmatrix}100\\0\end{bmatrix} =\begin{bmatrix}80\\20\end{bmatrix}.\]

The first output combines everyone arriving in A; the second combines everyone arriving in B.

6. Interpret One Row-Column Calculation

\[A_{n+1}=0.80A_n+0.30B_n,\qquad B_{n+1}=0.20A_n+0.70B_n.\]

Every next-state count includes contributions from both current states. Applying only the retention percentages would omit the members switching in from the other state.

7. Preserve the Total

Because the columns of M sum to 1,

\[A_{n+1}+B_{n+1}=A_n+B_n.\]

In this closed model, the total stays 100. A changed total signals an arithmetic error unless the context explicitly includes arrivals, departures, births, losses, or another outside flow.

8. Repeated Multiplication and Matrix Powers

\[\mathbf p_1=M\mathbf p_0,\qquad \mathbf p_2=M\mathbf p_1=M^2\mathbf p_0,\qquad \mathbf p_n=M^n\mathbf p_0.\]

The exponent n counts transition steps, not ordinary entrywise powers. Compute \(M^n\) through repeated matrix multiplication or appropriate technology.

9. Future-State Example

Step n\(A_n\)\(B_n\)
01000
18020
27030
36535
462.537.5
561.2538.75

The distribution approaches 60 in A and 40 in B. Decimal outputs are exact expected amounts in a proportional model; whether rounding is appropriate depends on the context.

10. Steady State

A steady-state vector s is unchanged by one more transition:

\[M\mathbf s=\mathbf s.\]

For the model above, let \(\mathbf s=\begin{bmatrix}A\\B\end{bmatrix}\) with \(A+B=100\). Then

\[0.8A+0.3B=A \quad\Longrightarrow\quad 0.2A=0.3B.\]

Combining \(A:B=3:2\) with the total 100 gives \(\mathbf s=\begin{bmatrix}60\\40\end{bmatrix}\).

11. Verify the Steady State

\[\begin{bmatrix}0.8&0.3\\0.2&0.7\end{bmatrix} \begin{bmatrix}60\\40\end{bmatrix} = \begin{bmatrix}48+12\\12+28\end{bmatrix} = \begin{bmatrix}60\\40\end{bmatrix}.\]

A numerical pattern that appears to settle is evidence; the equation \(M\mathbf s=\mathbf s\) is the exact verification.

12. Reconstruct a Past State

If M is invertible, then

\[\mathbf p_n=M\mathbf p_{n-1} \quad\Longrightarrow\quad \mathbf p_{n-1}=M^{-1}\mathbf p_n.\]

Here \(\det(M)=0.5\), so

\[M^{-1}= \begin{bmatrix}1.4&-0.6\\-0.4&1.6\end{bmatrix},\qquad M^{-1}\begin{bmatrix}80\\20\end{bmatrix} =\begin{bmatrix}100\\0\end{bmatrix}.\]

The inverse is a reconstruction tool, not itself a forward transition matrix; negative entries in \(M^{-1}\) can occur.

13. Validate a Reconstructed Past

Even when \(M^{-1}\) exists algebraically, the reconstructed vector must fit the context. Check that entries are nonnegative, totals are correct, units make sense, and applying M returns the known current state.

If the inverse produces a negative population, the proposed current state may not be reachable from a valid previous state under the model.

14. Assumptions and Model Limits

  • The same transition percentages apply at every step.
  • Each individual belongs to exactly one modeled state.
  • Transitions happen over equal-length intervals.
  • The system is closed unless outside flows are explicitly modeled.
  • Members transition independently enough for percentages to be meaningful.

A transition model may fail after a policy change, seasonal shift, capacity limit, or other event that changes the rates. Long-range predictions should be interpreted in light of these assumptions.

15. AP Workflow and Common Errors

  1. Define the state-vector order and units.
  2. Translate every retention and switching percentage into a decimal.
  3. Choose and label a row or column convention.
  4. Check that outgoing proportions total 1 for each source state.
  5. Multiply M by the current state for one future step.
  6. Use \(M^n\mathbf p_0\) for n steps and verify totals.
  7. Test a steady state with \(M\mathbf s=\mathbf s\), or use \(M^{-1}\) cautiously for a past state.
  • Do not place percentages without identifying their source and destination.
  • Do not mix a column-stochastic matrix with row-vector multiplication.
  • Do not square entries individually when calculating \(M^2\).
  • Do not round every intermediate state unnecessarily.
  • Do not declare a steady state from visual convergence alone.
  • Do not treat \(M^{-1}\) as a probability transition matrix.

Key Takeaways

  • Construct and iterate matrix models for transitions, populations, networks, and multivariable systems.
  • Core relationship: \(\mathbf p_{n+1}=M\mathbf p_n\)
  • Error check: Check whether rows or columns represent source states before interpreting probabilities.
Checkpoint · Topic 4.14
  1. Construct a column-stochastic transition matrix when 90% of A remains in A and 25% of B switches to A.
  2. Starting from \(\mathbf p_0=\begin{bmatrix}120\\80\end{bmatrix}\), calculate \(\mathbf p_1\) and interpret both entries.
  3. Use a matrix power to determine the state after four transitions.
  4. Find and verify the steady-state distribution when the total population is 200.
  5. Explain when \(M^{-1}\mathbf p_n\) can be used to reconstruct a previous state and list two contextual checks for the result.