A new slideshow on matrices has been added to the WolframAlpha series.
See the spreadsheets by Mike Hadden MatrixDeterimant and MatrixInverse, these enable you to check answers and see working out as well.
If you are looking for notes and examples on Matrices then the following are useful sources and appropriate for Advanced and degree level students.
Notes and examples
Chapter 9 on Matrices and Transformations from the CIMT Further Pure Mathematics A Level material,
Just the Maths,
The Math Centre
and The HELM Project. If you have not come across the HELM Project before, the project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include clear worked examples. You can see on the list that a very small number of titles are ‘not ready yet’; for the sake of completeness the complete set is hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.
David Smith’s site, The Maths Teacher has an extensive collection of videos to help you study Mathematics. GCSE (age 14-16), (though many of these resources would be helpful for younger students also-see Foundation as well as Higher) and A-Level (age 16-18) lessons are available. For each topic not only is a video available but also a transcript and exercises with solutions. This makes the site ideal for revision – you have the choice of perhaps just trying the exercises or if you feel you need more help you can watch the video – whatever is right for you.
There are many excellent sources of mathematics notes (all free) for students of all ages online. The notes page on this blog has a list of such resources.
To explore some of the resources on that list:
On mr barton maths.com you will find Craig Barton’s notes for students age 11 to 16 which older students might also find useful for revision. These notes are colourful and clear with carefully worked examples and are popular with students.
An extensive collection of very clear notes and other resources are available from the mathcentre which was developed by a group from the Universities of Loughborough, Leeds and Coventry, the Maths Stats and OR Network and the Educational Broadcasting Services Trust in 2003. There are many very clearly worked examples, also exercises (with answers).
A second mention for Coventry University with Just the Maths by Tony Hobson, another extensive collection of notes which I have often used used with older students. These notes are very useful for students in their last years at school and at university. There are numerous worked examples and also exercises to try. Looking for notes on second order differential equations recently it was the first site I tried and I wasn’t disappointed!
If you are trying any exercises you can obviously check the answers given in the notes; remember that you can also check your answers with WolframAlpha.
On the subject of WolframAlpha, there are several slideshows on the WolframAlpha page on this blog to help you get used to the syntax and a new one has just been added on differential equations.
Let’s suppose you are looking for some extra examples. You have probably checked all the usual websites, BBC Bitesize for example (also available on Facebook).
Some of my students were recently looking at examples like this and wanted some further problems:
Note the thinking behind this:
You should spot that the denominator is the difference of two squares, hence (x+5)(x-5)
If we are to have any hope of simplifying when we factorise the numerator, one of the factors must be (x+5) or (x-5), of those two it must be (x-5) or we would not be able to obtain the -19x term.
Factorise the numerator: (3x-4)(x-5)
A site perhaps not so well known by students is The Centre for Innovation in Mathematics Teaching. If you choose the GCSE course material and scroll down the page you will see all the pupil textbook chapters. (Also note all the other materials for students from 5 – 18!)
Chapter 10 on equations includes many useful algebra examples and exercises including problems of the type above (see worked example 1 on page 41 and the exercises on page 43). Note that each GCSE section has answers.
You should check any factorisation by multiplying out your answer. You could also check answers using WolframAlpha.
Note that you could enter the whole expression and simplify it – WolframAlpha will give you alternate forms; for this type of question you are require to give the simplest single fraction that you can.