Rucete ✏ AP Precalculus In a Nutshell
12. Matrices
This chapter introduces matrices as a powerful way to organize and manipulate data. You’ll explore matrix operations, inverses, determinants, transformations, and real-world applications such as modeling systems and predicting probabilities with Markov chains.
- What Is a Matrix?
- A matrix is a rectangular array of numbers organized into rows and columns.
- Size = number of rows × number of columns (e.g., 2 × 3).
- Elements are located by row then column: a₂₁ is in row 2, column 1.
- Real-world examples: calendars, spreadsheets, recipes.
- Matrix Operations
- Equality: Matrices must be same size with identical elements.
- Addition/Subtraction: Only for same-size matrices; add or subtract corresponding elements.
- Multiplication:
- Only possible if columns of first = rows of second.
- Multiply rows by columns and sum the products.
- Not commutative: AB ≠ BA
- Identity Matrix: Square matrix with 1’s on diagonal; AI = IA = A
- Inverse Matrix:
- Exists if det(A) ≠ 0; A⁻¹ × A = I
- 2 × 2 formula: A⁻¹ = (1/det) × [d −b; −c a]
- Determinant:
- For 2 × 2 matrix [a b; c d], det = ad − bc
- Used to find inverses and calculate areas
- Applications: Area and Geometry
- Area of parallelogram = |det(A)|, where A’s columns are vectors.
- If det = 0 → vectors are parallel (no area).
- Linear Transformations
- A transformation T maps vectors from Rⁿ to Rᵐ: T : Rⁿ → Rᵐ
- Linear if:
- T(u + v) = T(u) + T(v)
- T(ku) = kT(u)
- T(x) = Ax, where A is a matrix
- Rotation matrix: [cosθ −sinθ; sinθ cosθ]
- Dilation: Scales vectors; det tells area scaling
- Composition:
- If S and T are linear, then S ∘ T is linear
- Matrix of S ∘ T = product of matrices
- Inverse of Linear Transformations
- Transformation is invertible ⇔ its matrix is invertible (det ≠ 0)
- T⁻¹ maps image vector back to original
- Matrix of T⁻¹ = inverse of matrix of T
- Matrices as Functions
- Function f(x, y) = (2x + y, y, x − 3y) has a matrix A
- Use basis vectors to construct A
- Every matrix defines a linear transformation; converse is true if function is linear
- Markov Chains and Transition Matrices
- Markov chain: models state changes using probabilities
- Transition matrix: pᵢⱼ = probability of moving from state j to state i
- Each column in a Markov matrix sums to 1
- State vector: x = [x₁; x₂; ...] shows current probabilities
- Next state: xₙ₊₁ = P × xₙ
- Steady state: long-term result from repeated multiplication
- Example: track donations, library use, stock behavior
- Inverse of transition matrix can estimate previous states
In a Nutshell
Matrices are a versatile tool in algebra and modeling. From operations and transformations to predicting systems with Markov chains, they provide a powerful framework to solve complex, structured problems. Whether solving systems, rotating objects, or tracking probabilities, matrices bring structure and efficiency to advanced math.