Solving systems of linear equations using Gauss’s matrix method.

Dave Coulson, 2013

This is a 3x3 system.

Your task is to work out the values of x, y and z.

First, strip away the unnecessary symbols

... and put a box around the numbers.

Subtract.

Replace.

Subtract.

Replace.

Keep doing this so that the numbers turn to zero

Keep doing this so that the numbers turn to zero

... in no particular order.

... in no particular order.

There’s no ‘best’ way to do it.

There’s no ‘best’ way to do it.

Just be tidy.

Just be tidy.

And don’t make any mistakes.

And don’t make any mistakes.

Eventually...

Eventually...

...each column will have only one non-zero number left in it.

Put the unknowns back into position.

Put the unknowns back into position.

That’s the answer.

This procedure has been around for a very long time.

A Chinese scholar wrote a book on how to do this more than 2000 years ago.

Several other people independently rediscovered the method in later centuries.

Gauss popularised the method in the early 1800s and his name is attached to it as a result.

‘Gaussian elimination’ is taught in schools and universities all round the world today.

Computers are programmed to use the method on really large linear systems.

Using this same procedure, they can solve systems involving hundreds or even thousands of unknowns, in less time than it takes you to read this sentence.

But there are better ways, and still better ways are being discovered all the time.

Linear algebra is a big field of study today, and it’s as much about programming machines as it is about solving equations.

A typical space journey requires repeatedly solving equations using 7 variables: position in each of 3 directions, speed in each of 3 directions and a clock time.

The variables are put into a 7x7 matrix and identified using procedures similar to what you’ve just seen, advancing the clock time by maybe one second at a time across a journey that may take many months, so that hundreds of thousands of repetitive matrix calculations might be required to find out where the spacecraft ends up.

You can even calculate the motions of the planets using this sort of mathematics. Gauss used matrices to plot the motion of the asteroid Pallas, which is how he came to be interested in the subject in the first place.

You could say, therefore, that you discovered matrices because I discovered matrices, because my teachers discovered matrices, because somewhere back in that chain of teachers, Gauss demonstrated how useful the method could be, because he was interested in a tumbling rock in the sky some 50 million miles away.

The next question would be, who will you teach this subject to? And what will your student’s students discover, someday, using Gauss’s matrix method?

-David C, 2013