Modelling and control by Anthony Rossiter

# MATRICES and DETERMINANTS

These videos give an introduction to matrices beginning from definitions, simple operations and uses and then onto more elaborate algebra such as determinants and inverses. Matrices are a core mathematical tool that allow engineers to express complex problems in compact form thus allowing much easier manipulation and analysis.

Properties of determinants can be very useful and the insights often allow simplifications of otherwise computationally demanding problems. It is implicit that for many engineering topics, students have core competence in some mathematical topics such as polynomials, roots, complex numbers, matrices, exponentials, logarithms and Laplace. This section contains the following topics. Under each topic there are hardcopy (pdf) notes, a video talk through of key derivations with example problems and occasional tutorial questions for users to test themselves.

## Matrices 1 - what is a matrix?

Gives an introduction to matrices using simple examples and definitions.

## Matrices 2 - special matrices and equality.

A number of special matrices exist and are used in engineering problem solving. Some of these such as row and column vectors, square matrices, identity matrices, diagonal matrices, symmetric matrices and matric transpose are defined here for convenience of use later. The concept of matrix equality is also defined.

#### Quick test question

For matrices, which of the following statements is false.
A. In a symmetric matrix, Aij = Aji.
B. In an identity matrix, Aij = 1.
C. Two matrices A,B, are equal if Aij = Bij for all i,j.
D. For a matrix transpose, B=AT, then Aij = Bji.

## Matrices 3 - addition and subtraction.

Introduces the definition of matrix addition and subtraction and gives several numerical examples.

## Matrices 4 - use of MATLAB.

Gives readers a quite introduction to the entry and manipulation of matrices in the MATLAB environment. Demonstrates that the terminology and usage is equivalent to expected norms and thus it is an easy environment to use.

## Matrices 5 - multiplication.

Gives the definition of matrix multiplication. This is done in two ways. The first is a number of illustrations using real numbers and the second is a formal definition using a summation formulae. Shows how the dimension of a matrix product C=AB is linked to the dimensions of the two matrices, A,B.

## Matrices 6 - properties of multiplication.

It is important the users understand some key properties associated to matrix multiplication as this understanding can be used to simplify practical problems. This resources focuses on the following properties: (i) impact of multiplication by an identity matrix; (ii) concepts of a matrix product being zero even when neither matrix is non-zero; (iii) significance of the order of matrix multiplication when numerious matrices and multiplied together; (iv) concepts of commutativity and whether the existence of AB implies the existence of BA.

#### Quick test question

Which of the following is always true for matrices A,B,C? Assume dimensions allow the corresponding operations.
A. AB=BA.
B. A*(B*C)=(A*B)*C.
C. ATA=AAT
D. None of the above.

## Matrices 7 - engineering uses of multiplication

This resource is intended to help readers understand the potential uses of matrices for real problem solving. A few simple examples demonstrate how Matrix algebra is a convenient and compact way of represented real problems such as simultaneous equations, modelling and rotation/translation in files such as robotics.

## Matrices 8 - use of MATLAB

This resource demonstrates the use of MATLAB for matrix multiplication, data handling, simultaneous equations and for rotation/scaling with vector spaces.

## Matrices 9 - determinants for 2x2 matrices

Introduces the concept of a determinant and computation of this for 2 by 2 matrices. Demonstrates that scaling of vectors can be very different and that a small determinant indicates that for at least one vector direction the scaling can be very small.

Two obvious typos: (i) At 6min 4/16 should be 8/16. (ii) At 8 min F{1,2} should be +3 not -3.

## Matrices 10 - determinants for 3x3 matrices

Introduces the concept of a determinant for a 3x3 matrix. Requires definitions of MINORS and COFACTORS which are also given. Gives several worked examples but recognises that the general formulae is tedious to use and shortcuts are needed.

Includes a minor typo around 14.15 where a -4 is written as +4.

## Matrices 11 - determinants for large matrices

Introduces the general definition of determinant for large dimension matrices. Extends the definitions of MINORS and COFACTORS used in the definition of determinant. Demonstrates that the use of the general definition is very tedious in general and gives some numerical examples where shortcuts are possible. More shortcuts are in the following videos.

## Matrices 12 - shortcut methods for determinants

Previous videos introduced the concepts of a determinant but it was clear that in general these would be rather tedious to compute. This video introduces rules and shortcuts which allow much faster and easier computation. Here the focus is on the impact on the determinant of scaling rows, columns or all coefficients.

## Matrices 13 - shortcut methods for determinants continued

This video introduces two more properties of determinants. (1) Adding a multiple of any row or column to another row or column does not change the determinant and (2) if any row(column) is a multiple of any other row (column) then the determinant must be zero. Shows with examples how using these properties makes determinant computations much easier and quicker.

## Matrices 14 - shortcut methods for determinants continued

This video introduces a final properties of determinants which is that swapping rows and columns changes the sign (but not magnitude) of the determinant.

## Matrices 15 - determinants numerical examples

Gives a number of worked examples for evaluating determinants on pen and paper using properties and rules to simplify the algebra.

• A talk through video is on youtube.

## Matrices 16 - matrix inverse

These resources also fit logically into the section on simultaneous equations and thus are copied across directly from there.

### Simultaneous equations 13 – Gaussian elimination

Introduces Gaussian elimination methods for solving simultaneous equations and gives a number of worked examples to clarify the algorithm.

• summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.

### Simultaneous equations 14 – matrix inverse with Gaussian elimination

Shows that the same Gaussian elimination methods used for solving simultaneous equations can, with just a minor augmentation, be used to find a matrix inverse in a computationally efficient manner. Gives a number of worked examples to clarify the algorithm.

• A summary of key facts and derivations is given in the pdf file.
• A talk through video is on youtube.  