Lecture on Matrices

Introduction to Matrices

• Matrices are essential in various branches of mathematics and other fields like business, science, economics, sociology, etc.
• They simplify solving systems of linear equations, electronic spreadsheets, and represent physical operations like magnification and rotation.
• Used in cryptography and various scientific fields.

Definition of a Matrix

• An ordered rectangular array of numbers or functions.
• Elements are called entries.
• Example matrices provided:
  • A 3x2 matrix, B 3x3 matrix, C 2x3 matrix.

Order of a Matrix

• Defined as the number of rows (m) by number of columns (n): m x n matrix.
• Example: A is 3x2, B is 3x3, C is 2x3.

Types of Matrices

1. Column Matrix: One column only, e.g., [aij] m x 1.
2. Row Matrix: One row only, e.g., [aij] 1 x n.
3. Square Matrix: Number of rows equals number of columns, e.g., [aij] m x m.
4. Diagonal Matrix: Non-diagonal elements are zero.
5. Scalar Matrix: Diagonal elements are equal.
6. Identity Matrix: Diagonal elements are 1, others are zero.
7. Zero Matrix: All elements are zero.

Operations on Matrices

• Addition: Only defined for matrices of the same order. Element-wise addition.
• Scalar Multiplication: Multiplying every element by a scalar.
• Properties:
  • Commutative Law: A + B = B + A
  • Associative Law: (A + B) + C = A + (B + C)
  • Additive Identity: A + O = A
  • Additive Inverse: A + (-A) = O
• Matrix Multiplication: Defined when number of columns in A equals number of rows in B.

Properties of Matrix Multiplication

1. Associative: (AB)C = A(BC)
2. Distributive: A(B+C) = AB + AC
3. Multiplicative Identity: Exists for square matrices.

Transpose of a Matrix

• Interchanging rows and columns.
• Properties:
  • (A^T)^T = A
  • (A + B)^T = A^T + B^T
  • (AB)^T = B^T A^T

Symmetric and Skew-Symmetric Matrices

• Symmetric: A^T = A.
• Skew-Symmetric: A^T = -A (diagonal elements are zero).
• Any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix.

Invertible Matrices

• A square matrix A is invertible if there exists B such that AB = BA = I (identity matrix).
• Inverse is unique if it exists.

Summary

• Discusses various matrix types, properties, operations, and applications.
• Emphasizes the importance of matrices in simplifying mathematical computations and representing complex systems in a manageable form.