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AP Precalculus: Matrices Formulas & Properties

1. Matrix Vocabulary

  • Matrix: Rectangular array, dimensions \( m\times n \) (rows × columns)
  • Element: Entry \( a_{ij} \), row \(i\), column \(j\)
  • Identity Matrix \(I_n\): Square, ones on main diagonal, zeros elsewhere
  • Zero Matrix: All entries zero
  • Row/Column Matrix: \( 1\times n \) or \( m\times 1 \)
  • Square Matrix: \( n\times n \)

2. Matrix Operation Rules & Properties

  • Add/subtract \(A\) & \(B\) only if same dimensions
  • Multiplication \(A_{m\times n} B_{n\times p}\) defined if inner dimensions match (\(n\))
  • Not generally commutative: \( AB \neq BA \)
  • Identity: \( AI = IA = A \)
  • \( (AB)^T = B^T A^T \) (transpose rule)

3. Add, Subtract, and Scalar Multiplication

  • \((A+B)_{ij} = a_{ij} + b_{ij}\)
  • \((A-B)_{ij} = a_{ij} - b_{ij}\)
  • \((cA)_{ij} = c\cdot a_{ij}\) (scalar \(c\))
  • \(cA \pm dB = (c a_{ij}) \pm (d b_{ij})\)

4. Matrix Multiplication

  • \((AB)_{ij} = \sum_{k=1}^n a_{ik} b_{kj}\)
  • Result has dimensions \( m\times p \)

5. Simplifying Matrix Expressions

  • Apply operation rules above (watch dimensions!)
  • Combine like terms, distribute/scalar multiply before adding/subtracting

6. Fundamental Properties

  • \(A+B = B+A\)
  • \(A+(B+C) = (A+B)+C\)
  • \(A(B+C) = AB+AC\)
  • See also earlier operation rules

7. Determinant & Invertibility

  • 2×2: \( \det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad-bc \)
  • 3×3: Expand by minors or rule of Sarrus
  • Matrix \(A\) invertible if \( \det(A) \neq 0 \)

8. Matrix Inverse

  • 2×2: \( A^{-1} = \frac{1}{ad-bc} \begin{pmatrix}d & -b \\ -c & a\end{pmatrix} \)
  • 3×3: Use adjugate and determinant:
    • \( A^{-1} = \frac{1}{\det(A)} \operatorname{adj}(A) \)
  • \( AA^{-1} = A^{-1}A = I \)

9. Solve Matrix Equations

  • \( AX = B \implies X = A^{-1}B \) (if invertible)
  • Check \( \det(A) \neq 0 \)

10. Transformation Matrices

  • Transform a point: \( [x', y']^T = M [x, y]^T \)
  • Reflection over x-axis: \( \begin{pmatrix}1 & 0\\0 & -1\end{pmatrix} \)
  • Reflect y-axis: \( \begin{pmatrix}-1 & 0\\0 & 1\end{pmatrix} \)
  • Rotation by \( \theta \): \( \begin{pmatrix}\cos\theta & -\sin\theta\\\sin\theta & \cos\theta\end{pmatrix} \)
  • Vertex matrix: Columns are vertex coordinates