W10-11_Matrix Algebra

Introduction

Course: Fundamentals of Mathematics

Sessions: 10-11Topic: Matrix AlgebraInstructor: Erwan LamyAffiliation: ESCP Business SchoolLocations: Berlin, London, Madrid, Paris, Turin, Warsaw

Course Objectives

• Understand the concept of a matrix.

• Familiarity with special types of matrices.

• Learn matrix addition and scalar multiplication operations.

• Express a system of equations as a matrix equation using multiplication.

• Use matrix reduction techniques to solve linear systems.

• Explore the theory of homogeneous systems.

• Understand the concept of an inverse matrix.

Course Outline

Matrices (Chapter 6.1)

• Definition and Examples: Explore the definition of a matrix, recognize various examples, and identify dimensions.

• Methods: Identify and classify different examples of matrices based on size and entries.

Matrix Addition and Scalar Multiplication (Chapter 6.2)

• Methods:

  • Matrix Addition: For matrices A and B of the same size, add corresponding entries to obtain a new matrix.

  • Scalar Multiplication: Multiply each entry of matrix A by a scalar k to produce a new matrix kA.

Matrix Multiplication (Chapter 6.3)

• Methods:

  • Multiplication Process: Verify that the number of columns in A equals the number of rows in B.

  • Calculate Product: For entry (i, j), compute the sum of products of entries in the ith row of A and the jth column of B.

Solving Systems of Linear Equations

• Methods:

  • Matrix Representation: Express a system of equations as AX = B using coefficient and variable matrices.

  • Matrix Reduction: Use row operations to transform the system into row echelon form.

  • Inverse Matrix Method: Calculate the inverse of matrix A to find solutions by multiplying with the constant matrix.

Homogeneity and Number of Solutions (Chapter 6.5)

• Methods:

  • Analyze when a system has unique or infinite solutions depending on the rank and properties of the coefficient matrix.

Definition of Matrices

• A matrix is a rectangular array of numbers defined by its dimensions (m rows and n columns).

• The size of a matrix is denoted as m x n.

• An entry denoted as a_ij represents the value in the ith row and jth column of the matrix.

• Matrices A and B are equal if they share the same size and all corresponding entries are equal.

Examples of Matrices

• Given examples of various sizes:

  • 1x3 (row vector)

  • 3x2 matrix

  • 1x1 (scalar)

  • 3x5 matrix

Special Types of Matrices

• Square Matrix: An n x n matrix with equal number of rows and columns.

  • Example of a square matrix:[−5, −3, 1][1, −4, −1][0, −7, −4]

Types of Square Matrices

• Diagonal Matrix: A square matrix where all off-diagonal elements are zero.

  • Example:[−5, 0, 0][0, −1, 0][0, 0, 10]

• Upper Triangular Matrix: All entries below the diagonal are zero.

• Lower Triangular Matrix: All entries above the diagonal are zero.

• Zero Matrix: Denoted by O, an n x n zero matrix has all entries equal to 0:

  • Example: O₃ = [0,0,0;0,0,0;0,0,0]

Transpose of a Matrix

• The transpose (denoted A^T) of an m x n matrix A is an n x m matrix created by swapping rows and columns.

  • Example: A = [1, 2; 3, 4] implies A^T = [1, 3; 2, 4].

Matrix Addition and Scalar Multiplication

• Matrix Addition: A + B for two matrices A and B of the same size results in a matrix of the same size obtained by adding corresponding entries.

  • Method: For each entry in A and B, add the values directly.

• Scalar Multiplication: If A is an m x n matrix and k is any real number, then kA is a matrix obtained by multiplying every entry of A by k.

Properties of Matrix Addition

• Addition is defined only for matrices of the same size.

• Commutative Property: A + B = B + A

• Associative Property: A + (B + C) = (A + B) + C

Matrix Multiplication

• To multiply A (m x n) by B (n x p), the number of columns in A must equal the number of rows in B.

  • Method: The entry in position (i, j) of the product matrix AB is computed as:(AB)ij = sum of the products of the entries of the ith row of A and the jth column of B.

Properties of Matrix Multiplication

• Not commutative: AB ≠ BA

• Associative: A(BC) = (AB)C

• Distributive: A(B + C) = AB + AC

Solving Systems of Linear Equations

• Any system of linear equations can be represented in matrix form. For instance, a system represented as AX = B where A is the coefficient matrix and X is the variable matrix.

• Methods: Represent the system in matrix form, either via matrix reduction or applying the inverse matrix method.

Methods of Solving Linear Systems

• Matrix reduction and inverse matrix approaches are common methods for solving linear systems.

  • Matrix Reduction: Use row operations to reach row echelon form.

  • Inverse Matrix: Find the inverse of the coefficient matrix and multiply by the constant matrix.

Homogeneous Systems

• A system is homogeneous if all constant terms equal zero. For example, the system Ax = 0 is homogeneous.

• Method: Analyze the system for trivial and non-trivial solutions based on the rank of the matrix.

The Inverse Matrix

• A square matrix A has an inverse A^(-1) if and only if AA^(-1) = I (the identity matrix).

• Not all matrices are invertible.

Finding the Inverse

• To find A^(-1), augment A with the identity matrix and apply elementary row operations until the identity matrix appears on the left side of the augmented matrix.

Conditions for Solutions

• A linear system has a unique solution if its matrix is invertible. If not, it could have no solutions or infinitely many solutions.

Homogeneous System Solutions

• If the coefficient matrix has k non-zero rows, the solutions to the homogeneous system will depend on k:

  • < k: Infinitely many solutions.

  • = k: Unique trivial solution.

Non-Homogeneous System Solutions

• For non-homogeneous systems, the same k-row concept applies but it relates to existence of unique, no, or infinitely many solutions based on conditions evaluated through the Rouché-Capelli theorem.

Selected Examples

• Various examples illustrating concepts of matrix addition, multiplication, and linear system solutions can be useful in the application of these principles in practice.