Matrix Definition: A rectangular array of numbers or objects with defined operations (addition, multiplication). Elements are organized in rows and columns.
Matrix Basics
Size of a Matrix: Described by dimensions (m × n) where m = number of rows, n = number of columns.
Vectors:
Row vectors (1 × n matrix)
Column vectors (m × 1 matrix)
Special Types:
Square matrices (n × n)
Empty matrix (no rows/columns)
Infinite matrix (infinite rows/columns)
Matrix Notation and Operations
Notation: Matrices are denoted as A = [a_{ij}].
Conformability: Matrices are conformable for operations if dimensions allow it; e.g., same dimensions for addition.
Addition and Scalar Multiplication
Addition: A + B = [a{ij} + b{ij}] for conformable matrices A and B.
Scalar multiplication: kA = [k * a_{ij}].
Matrix Multiplication
Defined only when the number of columns in the first matrix equals the number of rows in the second. Resulting matrix has dimensions of the first matrix's rows and the second matrix's columns.
Special Matrices
Upper Triangular: All elements below the leading diagonal are zero.
Lower Triangular: All elements above the leading diagonal are zero.
Diagonal Matrix: Non-diagonal elements are zero.
Identity Matrix: All diagonal elements are 1, other elements are 0 (IA = A).
Determinants
For a 2 × 2 matrix A = [a b; c d], determinant is det(A) = ad - bc.
Used in solving linear systems (Cramer's Rule) and finding inverses.
Eigenvalues and Eigenvectors
Definition: λ is an eigenvalue of A if Av = λv for some nonzero vector v (the eigenvector).
Characteristic Polynomial: |A - λI| = 0.
Gaussian Elimination and Row Echelon Form
Method for solving linear equations; involves transforming matrices to simplify calculations.
Row echelon form has leading entries as 1, with all entries below as 0.
Inverse of a Matrix
Defined as A^(-1) = 1/det(A) * adj(A) where adj(A) is the transpose of cofactors of A.
Used to solve systems of equations: X = A^(-1)B.
Application in Linear Systems
Linear systems can be solved using inverses or elimination methods, determining solutions based on rows in echelon forms or using back substitution.