Section 1.3: Matrices and Matrix Operations
Introduction to Matrices
- Definition: A matrix is a rectangular array of numbers called entries.
- Examples and Sizes:
- 3x2 matrix: 3 rows and 2 columns.
- Row vector: 1 row, multiple columns (e.g., 1x4).
- Column vector: Multiple rows, 1 column (e.g., 3x1).
- Square matrix: Same number of rows and columns (e.g., 3x3).
- Notation:
- Uppercase letters for matrices, lowercase for scalars.
- Entry in row i, column j of matrix A denoted as A_ij.
- Compact notation: A = [A_ij] without specifying size if obvious.
Matrix Equality
- Two matrices are equal if they have the same size and corresponding entries are equal.
- Example: To determine equality between matrices, ensure all corresponding entries match (e.g., A = B if A_ij = B_ij for all i, j).
Matrix Addition and Subtraction
- Matrices can only be added or subtracted if they have the same size.
- Operation: Add/subtract corresponding entries.
- Example:
- A + B: Add corresponding entries.
- A - B: Subtract corresponding entries.
Scalar Multiplication
- Definition: Multiply each entry of a matrix by a scalar.
- Example: For matrices A, B, and C, find 2A, -B, and (1/3)C.
Matrix Multiplication
- Condition: Number of columns in the first matrix must equal the number of rows in the second matrix.
- Resulting Matrix: Size is determined by rows of the first matrix and columns of the second matrix.
- Process: Multiply each entry in a row by corresponding entry in a column, sum products.
- Example: Compute product of given matrices (e.g., A * B).*
Special Matrix Products
- Linear Combination: Expression of form C1A1 + C2A2 + ... + CrAr where C's are scalars.
- Column-Row Expansion: A matrix product expressed as sum of column and row products.
Transpose of a Matrix
- Transpose (A^T): Swap rows and columns of a matrix.
- Example: Find transposes of given matrices.
Trace of a Matrix
- Definition: Sum of diagonal entries of a square matrix.
- Condition: Only defined for square matrices.
- Example: Compute trace of given square matrices.
These notes summarize key concepts related to matrices and basic operations such as addition, subtraction, multiplication, scalar multiplication, equality, transpose, and trace, aiming to provide a comprehensive understanding of matrix operations.