AP PRECALCULUS — UNIT 4B · FUNCTIONS INVOLVING PARAMETERS, VECTORS, AND MATRICES
4.13B — Matrices as Functions
Notes — Powers and Iterated Application
💡 Learning Objectives (4.13.A Part 2)
By the end of this lesson you will be able to:
- Compute powers of a matrix A^n and interpret them as iterated transformations
- Apply a transformation repeatedly and track the result
- Recognize special patterns in matrix powers (idempotent, periodic, eventually zero)
- Use repeated transformations in modeling contexts
1. Matrix Powers
If A is a square matrix and n is a positive integer, A^n means applying A n times in succession. By the rules of matrix multiplication:
A² = A · A, A³ = A · A · A, …
By convention, A⁰ = I (the identity). And A⁻ⁿ = (A⁻¹)^n when A is invertible.
2. Computing A^n by Hand
For small n, just multiply. For larger n, look for PATTERNS — they are common with the matrices that come up in modeling.
📘 Example — A simple power
A = [[2, 0], [0, 3]] (a diagonal matrix).
- A² = [[4, 0], [0, 9]]
- A³ = [[8, 0], [0, 27]]
- Pattern: A^n = [[2^n, 0], [0, 3^n]]
📘 Example — A rotation
R = [[0, −1], [1, 0]] (90° rotation CCW).
- R² = [[−1, 0], [0, −1]] (180° rotation)
- R³ = [[0, 1], [−1, 0]] (270° rotation = 90° CW)
- R⁴ = I (back to start)
- Then R⁵ = R, R⁶ = R², … cyclic with period 4
3. Iterated Application to a Vector
Applying A repeatedly to a starting vector v₀ produces a SEQUENCE:
v₀, A v₀, A² v₀, A³ v₀, …
Each new vector is the previous vector multiplied by A. This generates a SEQUENCE OF POSITIONS — useful for modeling repeated processes.
📘 Example — Track a sequence
A = [[2, 0], [0, 1/2]], v₀ = [[1], [4]].
- v₁ = Av₀ = [[2], [2]]
- v₂ = Av₁ = [[4], [1]]
- v₃ = Av₂ = [[8], [1/2]]
- Pattern: x doubles each step, y halves each step
4. Special Patterns
💡 Recognizable Power Patterns
- Diagonal matrices: powers are easy — just power each diagonal entry
- Rotations: R^n is a rotation by n times the original angle; if angle is 2π/k, then R^k = I (periodic)
- Reflections: F² = I (reflecting twice returns to start); so F^n alternates between I and F
- Idempotent: A² = A (applying once or many times has the same effect after the first)
- Nilpotent: A^n = 0 for some n (eventually collapses everything to the origin)
5. When the Sequence Approaches a Limit
If the entries of A^n shrink to zero as n grows, then A^n v → 0 for every v. Geometrically, the iteration funnels every starting point toward the origin.
If A^n stabilizes (the entries approach finite values), the iteration approaches a steady state — common in population or economic models.
6. Long-Term Behavior — A Modeling Example
📘 Example — Two cities exchanging population
A = [[0.9, 0.2], [0.1, 0.8]] — each year, 10% of city 1 moves to city 2, and 20% of city 2 moves to city 1.
Starting populations [[1000], [500]]:
- Year 1: [[0.9·1000 + 0.2·500], [0.1·1000 + 0.8·500]] = [[1000], [500]]
- Year 2: same — already at steady state
- This particular A has a steady-state vector that does not change.
7. Compositions vs. Powers
Powers (A^n) and compositions (BA, ABC, etc.) work the same way mathematically but can be confused:
- A^n = applying the SAME transformation n times
- BA = applying A first, then B (different transformations)
- (AB)^n is NOT the same as A^n B^n in general (because matrices don't commute)
8. Summary
- A^n means applying A n times; A⁰ = I, A^(−n) = (A⁻¹)^n
- Diagonal matrices have easy powers
- Rotations have periodic powers (cycle back to I)
- Iterated application of A to a starting vector produces a sequence of states
- Long-term behavior: shrinking matrices ⇒ limit at origin; certain stable matrices ⇒ steady state