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Matrices and Transformations

Subject: Mathematics
Topic: 7
Cambridge Code: 0580

Introduction to Matrices​

Matrix - Rectangular array of numbers

Notation and Dimensions​

A=(a11a12a13a21a22a23)A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix}

  • Dimensions: 2 × 3 (rows × columns)
  • Element: aija_{ij} is in row i, column j

Types of Matrices​

Row matrix: 1 × n Column matrix: m × 1 Square matrix: n × n Identity matrix: Square with 1s on diagonal, 0s elsewhere Zero matrix: All elements are 0

Matrix Operations​

Addition and Subtraction​

Only possible for matrices of same dimensions

A+B=(a11+b11a12+b12a21+b21a22+b22)A + B = \begin{pmatrix} a_{11}+b_{11} & a_{12}+b_{12} \\ a_{21}+b_{21} & a_{22}+b_{22} \end{pmatrix}

Scalar Multiplication​

cA=(ca11ca12ca21ca22)cA = \begin{pmatrix} ca_{11} & ca_{12} \\ ca_{21} & ca_{22} \end{pmatrix}

Matrix Multiplication​

Note: Not commutative; AB ≠ BA generally

For A (m × n) and B (n × p) → AB is m × p

[AB]ij=∑k=1naikbkj[AB]_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}

Associative: (AB)C = A(BC) Distributive: A(B + C) = AB + AC

Transpose​

Transpose ATA^T - Swap rows and columns

A=(abcd),AT=(acbd)A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad A^T = \begin{pmatrix} a & c \\ b & d \end{pmatrix}

Properties:

  • (AT)T=A(A^T)^T = A
  • (AB)T=BTAT(AB)^T = B^T A^T
  • (A+B)T=AT+BT(A + B)^T = A^T + B^T

Determinant​

Determinant - Scalar value of square matrix

2 × 2 Determinant​

det⁡(A)=∣abcd∣=ad−bc\det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc

3 × 3 Determinant​

det⁡(A)=a(ei−fh)−b(di−fg)+c(dh−eg)\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Where A=(abcdefghi)A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}

Expansion along first row

Properties​

  • det⁡(AB)=det⁡(A)⋅det⁡(B)\det(AB) = \det(A) \cdot \det(B)
  • det⁡(AT)=det⁡(A)\det(A^T) = \det(A)
  • det⁡(cA)=cndet⁡(A)\det(cA) = c^n \det(A) (n = size)
  • Matrix is invertible iff det⁡(A)≠0\det(A) ≠ 0

Matrix Inverse​

Inverse A−1A^{-1} - Matrix satisfying AA−1=IAA^{-1} = I

2 × 2 Inverse​

For A=(abcd)A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}:

A−1=1ad−bc(d−b−ca)A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Only exists if det⁡(A)≠0\det(A) ≠ 0

Solving Systems Using Matrices​

AX=BAX = B X=A−1BX = A^{-1}B

Example: (2113)(xy)=(58)\begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 5 \\ 8 \end{pmatrix}

(xy)=(2113)−1(58)\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}^{-1} \begin{pmatrix} 5 \\ 8 \end{pmatrix}

Geometric Transformations​

Rotation​

Counterclockwise by angle θ: R=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

Reflection​

About x-axis: Fx=(100−1)F_x = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

About y-axis: Fy=(−1001)F_y = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}

About line y = x: Fy=x=(0110)F_{y=x} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

Scaling (Dilation)​

Scale by factor k: S=(k00k)S = \begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}

Shearing​

Horizontal shear: H=(1k01)H = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}

Composition​

Multiple transformations: Multiply matrices T=Tn⋅Tn−1⋅...⋅T1T = T_n \cdot T_{n-1} \cdot ... \cdot T_1

Apply right to left

Eigenvectors and Eigenvalues​

Eigenvalue Equation​

Av=λvA\mathbf{v} = λ\mathbf{v}

Where:

  • λ = eigenvalue (scalar)
  • v = eigenvector (non-zero)

Characteristic Equation​

det⁡(A−λI)=0\det(A - λI) = 0

Solve for λ (eigenvalues)

Finding Eigenvectors​

For each eigenvalue λ, solve: (A−λI)v=0(A - λI)\mathbf{v} = \mathbf{0}

Example​

For A=(3113)A = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}:

det⁡(3−λ113−λ)=(3−λ)2−1=0\det\begin{pmatrix} 3-λ & 1 \\ 1 & 3-λ \end{pmatrix} = (3-λ)^2 - 1 = 0

λ2−6λ+8=0λ^2 - 6λ + 8 = 0 (λ−2)(λ−4)=0(λ - 2)(λ - 4) = 0

Eigenvalues: λ = 2, λ = 4

Diagonalization​

Diagonalization - Expressing A in form PDP^(-1)

A=PDP−1A = PDP^{-1}

Where:

  • D = diagonal matrix of eigenvalues
  • P = matrix of eigenvectors

Powers of Matrices​

An=PDnP−1A^n = PD^nP^{-1}

Easier to compute when diagonalized

Applications​

Computer Graphics​

Transformations of images

Engineering​

Systems of linear equations

Population Models​

Transition matrices for population dynamics

Markov Chains​

Transition probabilities between states

Key Points​

  1. Matrix operations: Addition, subtraction, multiplication
  2. Determinant measures invertibility
  3. Inverse exists iff determinant ≠ 0
  4. Matrices represent geometric transformations
  5. Rotation, reflection, scaling matrices
  6. Composition of transformations: Matrix multiplication
  7. Eigenvalues found from characteristic equation
  8. Eigenvectors are invariant direction under transformation
  9. Diagonalization simplifies matrix powers
  10. Applications in graphics, engineering, population models

Practice Questions​

  1. Perform matrix operations
  2. Calculate determinants
  3. Find matrix inverses
  4. Solve systems using matrices
  5. Apply transformation matrices
  6. Compose transformations
  7. Find eigenvalues
  8. Find eigenvectors
  9. Diagonalize matrices
  10. Apply to real problems

Revision Tips​

  • Understand matrix dimensions for operations
  • Memorize 2×2 inverse formula
  • Know transformation matrix forms
  • Practice matrix multiplication
  • Understand eigenvalue significance
  • Connect to geometric transformations
  • Work with systems of equations
  • Apply to practical applications
  • Verify results check