Matrices, Determinants, and Systems of Linear Equations

Do you want to learn how to work with matrices and their properties, operations, and applications? Do you want to understand the concepts of determinants, eigenvalues, eigenvectors, diagonalization, quadratic and canonical forms? Do you want to master the skills of solving systems of linear equations, finding inverses, and computing matrix functions using different methods and tools? If you answered yes to any of these questions, then this course is for you! This course covers the fundamentals of matrix theory and its applications in mathematics and science. You will learn how to: - Define and classify matrices and their special types, such as symmetric, orthogonal, diagonal, and identity matrices - Perform matrix addition, subtraction, multiplication, and scalar multiplication using the algebraic properties of matrices - Find the transpose, conjugate, and adjoint of a matrix and use them to simplify matrix operations and expressions - Perform elementary row and column transformations on matrices and use them to find the row echelon form, reduced row echelon form, rank, and nullity of a matrix - Find the minors, cofactors, and determinants of matrices and use them to calculate the area, volume, and orientation of geometrical figures - Find the inverse of a matrix using the adjoint method or the row operations method and use it to solve systems of linear equations - Find the eigenvalues and eigenvectors of a matrix using the characteristic polynomial and the Cayley-Hamilton theorem and use them to analyze the behavior and stability of dynamical systems - Diagonalize a matrix using the eigenvalues and eigenvectors and use it to compute matrix functions, such as polynomials, exponentials, sines, and cosines of matrices - Find the quadratic and canonical forms of an equation using the coefficient matrix and the transformation matrix and use them to classify and graph conics and quadrics - Use computer software, such as MS-Excel, MATLAB, and Python, to perform matrix operations and computations efficiently and accurately This course is suitable for anyone who wants to learn or review the basics of matrix theory and its applications. It is especially useful for students and professionals in algebra, calculus, differential equations, linear programming, optimization, cryptography, computer graphics, data science, machine learning, and other related fields. By the end of this course, you will have a firm grasp of the theory and applications of matrices and determinants. You will also be able to apply the knowledge and skills you gain to solve real-world problems and challenges that involve matrices. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

26 hrs

$ 10.00

Payment required for enrolment

Enrolment valid for 12 months

This course is also part of the following learning tracks. You may join a track to gain comprehensive knowledge across related courses.

MTH 202: Mathematical Methods II

Comprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

CHE 305: Engineering Analysis I

Advanced engineering mathematics covering solid analytical geometry, integrals, scalar and vector fields, matrices and determinants and complex variables. Curated for third-year students of engineering at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Course Chapters

Introduction

Meaning of matrices, types of matrices, and other descriptive terminologies.

Chapter lessons

1.Welcome8:27

Welcome to the course and outline of course.

2.Definition21:17

Meaning of a matrix; order, row, column and field of a matrix.

3.Some special matrices (1)9:32

Row, column and null matrices.

4.Some special matrices (2)21:29

Square, diagonal, scalar and identity (unit) matrices.

5.Some special matrices (3)20:56

Upper and lower triangular matrices, banded matrices, tridiagonal matrices.

6.Some special matrices (4)25:47

Sparse and dense matrices, diagonally-dominant matrices.

Algebra of Matrices

Addition of matrices, multiplication of matrices by scalars, multiplication of matrices by matrices, etc.

Chapter lessons

1.Equality of matrices6:21

When are two matrices said to be equal?

2.Matrix addition8:50

How to add or subtract two or more matrices.

3.Properties of matrix addition6:53

Commutativity, associativity, identity and inverse in the operation of addition of matrices.

4.Scalar multiplication7:34

How to multiply a matrix by a number.

5.Properties of scalar multiplication7:22

Properties of scalar multiplication of matrices - distributivity, associativity, etc.

6.Matrix multiplication21:24

How to multiply a matrix by another matrix.

7.Properties of matrix multiplication10:51

Non-commutativity, associativity, distributivity over addition, identity and inverse elements for the operation of multiplication of matrices.

8.Worked examples (1)32:31

Worked examples on algebra of matrices.

9.Worked examples (2)10:50

More worked examples on algebra of matrices.

Transposition of Matrices

Transposition of matrices and properties of matrix transposes.

Chapter lessons

1.Definition6:43

Meaning and notations for the transpose of a matrix.

2.More special matrices (1)22:58

Symmetric and skew-symmetric matrices.

3.More special matrices (2)44:07

Orthogonal matrices.

4.Conjugates and conjugate transposes11:36

Meaning of conjugates and conjugate transposes for matrices with complex entries.

5.More special matrices (3)23:08

Hermitian and skew-Hermitian (anti-Hermitian) matrices.

6.More special matrices (4)19:34

Unitary matrices.

7.Properties of the transpose18:49

Properties of matrix transposes.

8.More worked examples (1)6:40

More worked examples on transposes and properties of matrix transposes.

9.More worked examples (2)18:42

More worked examples on transposes and properties of matrix transposes.

Elementary Transformations

Elementary row and column operations on matrices; row echelon and reduced row echelon forms; rank and nullity of matrices.

Chapter lessons

1.Introduction27:40

Meaning of elementary transformations - elementary row and column operations.

2.Equivalent matrices5:45

Meaning of equivalence, row equivalence and column equivalence of matrices.

3.Elementary matrices16:52

Meaning, notations and examples of elementary matrices.

4.Theorems (1)23:50

Some theorems on elementary transformations - row and column operations.

5.Theorems (2)24:33

Some theorems on elementary matrices.

6.Theorems (3)8:41

Some theorems on equivalent matrices.

7.Row echelon form (1)13:38

Meaning of a row echelon form of a matrix.

8.Row echelon form (2)23:06

Reduction of a matrix to a row echelon form.

9.Reduced row echelon form (1)10:56

Meaning of the reduced row echelon form of a matrix.

10.Reduced row echelon form (2)27:58

Reduction of a matrix to its reduced row echelon form.

11.Rank and nullity of a matrix4:58

Meaning of rank and nullity of a matrix.

12.More worked examples (1)24:34

More worked examples on elementary row operations.

13.More worked examples (2)19:08

More worked examples on elementary row operations.

Determinants

Meaning, operations and properties of determinants.

Chapter lessons

1.Introduction6:41

Meaning and notations of determinants, singular and non-singular matrices.

2.Determinants of orders 1 and 26:12

Computing determinants of orders 1 and 2.

3.Signs, minors and cofactors18:41

Meaning of sign factor, minor and cofactor of an element in a matrix.

4.Determinants of any order34:27

Formal definition (Laplace expansion formula) of determinants, and how to evaluate determinants of any order.

5.Properties of determinants (1)11:30

Determinants of matrix transposes and products, similar matrices, etc.

6.Properties of determinants (2)9:06

Properties of determinants involving zero or identical rows (or columns), triangular and diagonal matrices.

7.Properties of determinants (3)23:50

Properties of determinants involving elementary row (or column) operations.

8.Properties of determinants (4)12:50

Properties of determinants involving sums, derivatives and integrals.

9.More worked examples (1)26:33

Worked examples on evaluating determinants by row and column operations.

10.More worked examples (2)25:21

More worked examples on evaluating determinants by row and column operations.

11.More worked examples (3)14:19

More worked examples on evaluating determinants by row and column operations.

12.Cramer's rule19:13

How to solve a square system of linear equations using determinants - Cramer's rule.

13.More worked examples (4)26:17

More worked examples on solutions of square systems of linear equations using Cramer's rule.

Matrix Inverses

Meaning, operations and properties of matrix inverses.

Chapter lessons

1.Adjoints8:01

Definition of the adjoint of a matrix.

2.Properties of adjoints9:22

Properties of the adjoint of a matrix and their applications.

3.Matrix inverse18:01

Definition of the inverse element and the inverse of a matrix.

4.Properties of matrix inverses (1)14:16

Properties of matrix inverses and their applications - uniqueness, inverses of products, transposes and scalar multiples.

5.Properties of matrix inverses (2)26:28

Properties of matrix inverses and their applications - elementary row operations on an invertible matrix to give the identity matrix.

6.Properties of matrix inverses (3)27:41

Properties of matrix inverses and their applications - solutions of square systems of linear equations.

7.Properties of matrix inverses (4)7:53

Properties of matrix inverses and their applications - implications on rank, determinant, homogeneous square systems of linear equations, etc.

8.More special matrices7:12

Orthogonal, singular, non-singular, invertible and non-invertible matrices.

9.Similar matrices19:16

When are two matrices said to be similar?

10.More worked examples (1)27:13

More worked examples on computing matrix inverses.

11.More worked examples (2)22:52

More worked examples on computing matrix inverses.

Systems of Linear Equations

Application of matrices, determinants and inverses in solving systems of linear equations, consistency of systems of linear equations and applicable theorems.

Chapter lessons

1.Definition28:07

Meaning, matrix representation and homogeneity of systems of linear equations

2.Solutions15:48

Meaning and kinds of solutions of systems of linear equations; meaning of consistency of systems of linear equations.

3.Equivalent systems42:00

When are two systems of linear equations equivalent? What is the impact of elementary row operations on the systems of equations?

4.Consistency28:20

How to detect inconsistency in systems of linear equations using elementary row operations - how the ranks of coefficient and augmented matrices tell inconsistency.

5.Nonhomogeneous systems20:00

Conditions for existence of different forms of solutions for nonhomogeneous systems of linear equations.

6.Homogeneous systems6:07

Conditions for existence of different forms of solutions for homogeneous systems of linear equations.

7.Pivots and free variables27:36

How to determine what variables to parameterize in reporting infinitely many solutions for linear systems of equations.

8.More worked examples (1)16:04

More worked examples on solutions of general systems of linear equations.

9.More worked examples (2)21:40

More worked examples on solutions of general systems of linear equations.

Computer-Aided Handling

Manipulating matrices with Microsoft Excel (Google Sheets) software.

Chapter lessons

1.MS-Excel (Google Sheets)13:50

Manipulating matrices with MS-Excel (Google Sheets) - algebra, transposes, determinants and inverse of matrices.

2.Worked examples (1)8:41

Worked examples on manipulating matrices with MS-Excel (Google Sheets) - solution of a square system of linear equations.