Chapter one

This chapter is on the theme of supporting mathematics that are needed for engineering problem solving, analysis, design and evaluation.

Matrices are a core mathematical tool that allow engineers to express complex problems in compact form, allowing much easier manipulation and analysis. Properties of determinants can be very useful. The insights often allow simplifications of otherwise computationally demanding problems.

1. What is a matrix?

This section gives an introduction to matrices using simple examples and definitions.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 449KB).

2. Special matrices and equality

A number of special matrices exist and are used in engineering problem solving. Some of these are row and column vectors, square matrices, identity matrices, diagonal matrices, symmetric matrices and matrix transpose.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 432KB).

3. Addition and subtraction

This section introduces the definition of matrix addition and subtraction and gives several numerical examples.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 288KB).

4. Use of MATLAB

An introduction to the entry and manipulation of matrices in the MATLAB environment. This demonstrates that the terminology and usage is equivalent to expected norms. Therefore, it is an easy environment to use.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 1.0MB). View the MATLAB m-files, matrixentry.m.

5. Multiplication

The definition of matrix multiplication. This section includes a number of illustrations using real numbers and a formal definition using a summation formulae. It shows how the dimension of a matrix product C=AB is linked to the dimensions of the two matrices, A,B.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 292KB).

6. Properties of multiplication

This section includes information on the impact of multiplication by an identity matrix, the matrix product being zero even when neither matrix is non-zero, the significance of the order of matrix multiplication and whether the existence of AB implies the existence of BA.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 362KB).

7. Engineering uses of multiplication

Information to understand the potential uses of matrices for problem solving. This section demonstrates how Matrix algebra is a convenient and compact way to represent problems such as modelling, simultaneous equations and rotation/translation.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 410KB).

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.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 1.1MB). View the MATLAB m-files, matrixmult.m and simulteqn.n.

9. Determinants for 2x2 matrices

This section introduces the concept of a determinant. It also covers computation of this for 2 by 2 matrices and some implications/interpretations.

There is a typing error at 6 minutes, 4/16 should be 8/16. There is a typing error at 8 minutes, F{1,2} should be +3 not -3.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 321KB).

10. Determinants for 3x3 matrices

This section explains determinants for a 3x3 matrix and gives definitions of minors and cofactors. There are several worked examples. The general formulae is tedious to use and shortcuts are needed.

There is a typing error at 14 minutes 15 seconds, a -4 is written as +4.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 345KB).

11. Determinants for large matrices

This section explains determinants for large dimension matrices. It gives extended definitions of minors and cofactors. The general definition is very tedious, so it gives some numerical examples where shortcuts are possible.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 483KB).

12. Shortcut methods for determinants

This video introduces rules and shortcuts which allow much faster and easier computation. The focus is on the impact on the determinant of scaling rows, columns or all coefficients.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 699KB).

13. Shortcut methods for determinants continued

Exploiting properties of matrices to make computations easier. This involves adding a multiple of any row or column to another row or column. Also, showing how a row or column is a multiple of any other row or column.

A talk through video is on YouTube. View the PowerPoint slides (PDF, 889KB).

14. Shortcut methods for determinants continued

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

A talk through video is on YouTube. View the PowerPoint slides (PDF, 559KB).

15. Determinants numerical examples

This video 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.

16. Matrix inverse

These resources also fit logically into the section on simultaneous equations. Therefore, you may choose to look at the parts on Gaussian elimination.