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Prove that matrix multiplication is not commutative.

At GCSE level, proof questions are relatively rare and largely will all require a similar sort of approach. The difference with A Level is that the syllabus contains more than one method of proof. You will be expected to select and apply the appropriate method to sometimes unseen questions. This can be tricky!

For this reason, my personal advice for working with proof questions is to not rely too much on memory. It is not possible to memorise how to prove every single possible fact within mathematics, and if even it were possible it definitely would not be very efficient! Instead, my approach is to develop a good structured thought process that I can rely on using in any such question.

Firstly, ask yourself what needs to be proved. Do you understand all of the terms used? In this example, the word “commutative” may be unfamiliar. I tend to remember it by thinking about the word “commute” – to move around. If multiplication is commutative then it means the answer is ALWAYS the same when we switch the order it is carried out in. For example, multiplication of real numbers is commutative since whether we write a*b or b*a the answer is always the same. (I.e. 3*4 = 12 and 4*3 = 12).

So to show that matrix multiplication is NOT commutative we simply need to give one example where this is not the case. This is called disproof by counterexample.

Now we have effectively re-worded the problem to make it easier for us to tackle. We now know that we need to come up with a single example where multiplying two matrices one way is not the same as doing it the other.

It just so happens that this is nearly always the case, so we can simply chose two matrices (keep it simple – 2X2 matrices will do nicely) and do the calculations. Let’s call one of them A and the other B. We thus need to show that AB is not equal to BA. Using our examples, calculate directly AB (giving a 2X2 matrix answer) and then do the same for BA (giving a different answer to the last). If the answers happen to be the same (not very likely), then you can simply change one of the matrices slightly and try again. Since the answers produced are not the same, clearly the order of multiplication affects things! This is the counterexample we wanted and so we have proved the statement in the question.

Another good way to do this is to write down ANY 3X2 matrix (call this C) and ANY 2X2 matrix (call this D). We are able to calculate CD as normal via matrix multiplication, however if we try to calculate DC we see that we are unable to do so – the dimensions mean they cannot be multiplied! This is also proof of the statement and saves on time in the exam since less work is required.

Samuel F. A Level Further Mathematics  tutor, GCSE Maths tutor, A Lev...

2 years ago

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