Overview
This lecture covers core concepts of matrices and determinants, their properties, operations, and applications, including solving systems of equations.
Matrix Basics
- A matrix is a rectangular array of numbers arranged in rows and columns.
- The order of a matrix is given by "m x n", where m = rows, n = columns.
- Elements are denoted by aᵢⱼ, meaning i-th row and j-th column.
- A square matrix has equal number of rows and columns (e.g., 2x2, 3x3).
Types and Properties of Matrices
- Equality: Matrices are equal if they have the same order and all corresponding elements are equal.
- Scalar multiplication: Multiply each element by a scalar (number).
- Identity matrix: A square matrix with ones on the diagonal and zeros elsewhere.
- Transpose: Interchange rows and columns of a matrix.
- Symmetric matrix: A = Aᵗ (transpose).
- Skew-symmetric matrix: A = –Aᵗ.
Matrix Operations
- Matrix addition/subtraction: Only possible if matrices have the same order.
- Matrix multiplication: Possible if number of columns in first = rows in second; result has order (rows of first) x (columns of second).
- (AB)ᵗ = BᵗAᵗ.
- Any square matrix can be written as the sum of a symmetric and skew-symmetric matrix: ½(A + Aᵗ) + ½(A – Aᵗ).
Determinants
- Only defined for square matrices.
- Determinant of 2x2 matrix: |a b| = ad – bc.
|c d|
- For 3x3 matrix, expand using rule of Sarrus or cofactors.
- det(kA) = kⁿ det(A) for an n x n matrix.
- det(A) = det(Aᵗ).
- Singular matrix: Determinant = 0; Non-singular: Determinant ≠ 0.
Minors, Cofactors, and Adjoint
- Minor: Determinant of submatrix after removing i-th row and j-th column.
- Cofactor: Minor multiplied by (–1)^(i+j).
- Adjoint: Transpose of the matrix of cofactors.
Inverse of a Matrix
- A⁻¹ = 1/det(A) × adj(A), if det(A) ≠ 0.
- A × A⁻¹ = Identity matrix.
Applications
- Area of triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃): ½ × |x₁ y₁ 1|
|x₂ y₂ 1|
|x₃ y₃ 1|
- Determinants can be used to test for collinearity of points.
- To solve system AX = B: X = A⁻¹B if A is invertible.
Key Terms & Definitions
- Matrix — Rectangular arrangement of numbers in rows and columns.
- Order — Dimensions of a matrix (rows x columns).
- Determinant — A unique number associated with square matrices.
- Minor — Determinant of a smaller matrix after deleting a row and column.
- Cofactor — Minor with sign, used in determinant expansion.
- Adjoint — Transpose of the matrix of cofactors.
- Inverse Matrix — A⁻¹ such that AA⁻¹ = I.
- Singular Matrix — Matrix with determinant zero.
Action Items / Next Steps
- Practice problems on matrix operations and properties.
- Calculate determinants for 2x2 and 3x3 matrices.
- Find the inverse for given matrices.
- Solve systems of equations using matrices and determinants.