CHAPTER 1
Matrices and How to Manipulate Them
What Is a Matrix?
You are at home in the evening; there is nothing good on television; and you are at a loose end. There are three possibilities open to you: to go out to the pub, to go out to the theatre, or to stay at home and invite some friends round for a game of cards. In order to weigh up the comparative advantages and disadvantages of these three alternatives, you decide to put certain basic facts about each of them down on paper. And, being an orderly, methodical type you put them down in the form of a table, like this:
Motoring Admission Liquor Crisps
(miles) charge (£s) (pints) (packets)
Go to pub: 3 1 4 3
Go to theatre: 2 3 1 1
Cards at home: 0 0 12 9
If you went to the pub, you would have to take your car out and drive three miles. It would cost you £1 to get in, since there is a special entertainment on there tonight, and you would also have to pay for the four pints of liquor and three packets of crisps which you calculate that you would consume on the premises. If you went to the theatre, it would cost you £3 to get in, but as compared with the pub you would save a little on motoring costs and quite a lot on liquor and crisps. If you stayed at home and asked some friends round for cards, you would have to provide a comparatively large quantity of liquor and crisps, but to compensate for this you would not have to take the car out and there would not of course be any admission charge.
Suppose now that just for fun you extracted the array of numbers from this table and put a pair of large square brackets around them, like this:
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No doubt much to your surprise, you would then have succeeded in constructing a matrix, which is simply a rectangular array of numbers. The individual numbers in a matrix are called its components or elements. This particular matrix, since it has three rows and four columns, is said to be a 3 × 4 matrix. If you had considered only two alternative courses of action – omitting, say, the 'Go to theatre' possibility – the matrix you constructed would have looked like this:
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This matrix has two rows and four columns, and is therefore said to be a 2 × 4 matrix. If on the other hand you had retained all three rows but omitted one of the columns, your matrix would have been a square one with three rows and three columns – that is, a 3 × 3 matrix.
Suppose now that you had seriously considered only one possible course of action – going to the pub, say. Your table would then have consisted of a 1 × 4 matrix – that is, a single row of four numbers:
[3 1 4 3].
An ordered collection of numbers written in a single row like this is a special (and very important) kind of matrix which is called a row vector.
Suppose finally that you had been interested only in the different quantities of liquor consumption associated with the three alternative courses of action. Your table would then have consisted of a 3 × 1 matrix – a single column of three numbers:
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An ordered collection of numbers written in a single column like this is another special (and equally important) kind of matrix which is called a column vector.
Matrices in the Social Sciences
The next question, of course, is why matrices are important in the social sciences, and why so many textbooks spend so much time instructing you how to play around with them. What is it all about?
A short answer is that the social sciences are often concerned with unravelling complex interrelationships of various kinds, and that it is often extremely convenient and illuminating to put these interrelationships down on paper in matrix form.
In economics, for example, we may be interested in the implications of the fact that some of the things which an industry produces (that is, its 'output') may be used as ingredients (that is, as 'inputs') in the production of other things – or even in their own production. A large part of the electricity produced in this country, for example, is not consumed directly by you and me, but is used as an input in the production of things like corn, machines, clothes and so on – and of electricity itself. Imagine, then, a very simple economy where there are only three industries, which we shall imaginatively call A, B and C. Industry A produces 300 units of its particular product – tons of steel, kilowatt hours of electricity, or whatever we suppose it to be – every year. It sells 50 of these 300 units to itself (as it were) for use as an input in its own production process; it sells 100 units to industry B and 50 to industry C for use as inputs in their production processes; and the remaining 100 units are sold to final consumers like you and me. Industry B produces 150 units, 70 of which go to itself, 25 to A, 5 to C, and the remaining 50 to final consumers. Industry C produces 180 units, 60 of which go to itself, 30 to A, 10 to B, and 80 to final consumers.
The way in which the total outputs of the three industries are disposed of can be very conveniently set out in the form of a simple matrix like this:
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One of the advantages of setting out the facts in this way is that two interrelated aspects of the overall situation are presented to us at one and the same time: the three rows tell us where each industry's output goes to, and the first three columns tell us where each industry's physical inputs come from.
Another type of matrix which often crops up in the social sciences is one which sets out the gains and losses accruing from some kind of 'game' which two (or more) participants are supposed to be playing. Suppose, for example, that two persons, Tom and Jerry, find themselves in some sort of conflict situation in which they are obliged to choose (independently of one another) between several alternative courses of action, and in which the final outcome – the gain or loss for each 'player' – depends upon the particular combination of choices which they make. Tom and Jerry, let us say, are two rival candidates for political office. At a certain stage in the election campaign a crisis arises, in which Tom has to choose between two possible...
