Lecture Notes on Matrices

Summary

In this lecture, we focused on the basics of matrices, defining them as arrays of numbers arranged in rows and columns. Key topics included understanding the order of a matrix, identifying specific elements in a matrix, and operations on matrices including addition, subtraction, and scalar multiplication.

Matrix Basics

• Definition: A matrix is an array of numbers arranged in rows (horizontal) and columns (vertical).

• Order of a Matrix: The order of a matrix is determined by the number of rows and columns it contains. It is expressed as "rows × columns".

  Examples

  • Matrix A has an order of 2×3 (2 rows and 3 columns: [2, 7, -4], [6, 3, 5]).
  • Matrix B has an order of 3×4 (3 rows and 4 columns: [4, 3, 7, -2], [5, 6, -4, 9], [-3, 8, 1, -7]).

Elements of a Matrix

• Identifying Elements: Elements are referred to using a subscript notation, where the first number is the row index and the second number is the column index (e.g., a₂₃ refers to the element in the 2nd row and 3rd column of Matrix A).

  Examples

  • In Matrix A, a₂₃ = 5, a₁₂ = 7, a₂₁ = 6.
  • In Matrix B, b₁₁ = 4, b₂₃ = -4, b₁₄ = -2, b₃₄ = -7.

Specific Matrix Examples

• Matrix C: 2×2 and square matrix (rows = columns).
• Matrix D: 3×2 matrix.
• Matrix E: 1×1 and square matrix.
• Matrix F: 1×4 matrix.
• Matrix G: 3×3 and square matrix.
• Matrix H: 2×4 matrix.

Matrix Operations

Addition

• Only possible between matrices of the same order.

• Elements are added corresponding to their positions.

  Example

  • Adding Matrix A and B (if they have the same order):
    • Resultant matrix elements: (a₁₁ + b₁₁, a₁₂ + b₁₂, ..., aₙₘ + bₙₘ).

Scalar Multiplication

• Every element of the matrix is multiplied by the scalar value.

  Example

  • Multiplying Matrix A by 4:
    • Resultant matrix: [8, 12, 20, -16] (from [2, 3, 5, -4] × 4).

Subtraction

• Similar to addition, corresponding elements are subtracted.

  Example

  • Subtracting Matrix B from A:
    • Resultant matrix elements are calculated as (a₁₁ - b₁₁, a₁₂ - b₁₂, ..., aₙₘ - bₙₘ).

Summary

This class covered the foundational elements of matrices from identification, operations, and specific matrix operations, all integral to understanding and working with matrices in algebra and beyond.