122 CHAPTER 10

Thus Z =

4

2

, and indeed you can check that x = 4, y = 2solvesthe

original system (10.1)–(10.2).

J ust by glancing at them, who would have thought that the num-

ber 19 was lurking in the system of equations (10.1)–(10.2)? But if

you change the constants on the right-hand side of the equations,

you are likely to ﬁnd 19’s popping up in the denominators of the

solutions. This is related to the fact that the determinant of the

matrix A is 19 (see chapter 11).

There is a complete theory of this and it gives you very efﬁcient

ways to solve many simultaneous equations in many unknowns, as

long as none of the exponents is greater than 1. If you have many

equations and unknowns, you should probably use a computer to

help you, but the computer will be using matrices. At the end of the

next chapter we will do a 3-by-3 example.

Digression: Graeco-Latin Squares

Matrices with entries that are not numbers can still have uses.

For example, a game from the 1950s consisted of a 6-by-6 grid of

squares and 36 pieces. The plastic pieces were of six different colors

and six different shapes, so that each piece was different from all

the others. The challenge was to arrange the pieces on the grid in

such a way that no row or column contained two pieces of the same

shape or same color. The manufacturer offered a prize of $1000 for

the ﬁrst solution. Some people tried this for quite a while.

In fact, the eighteenth-century mathematician Leonhard Euler

had already publicized this problem, which has a long subsequent

history. Instead of plastic pieces, he worked with soldiers, each of

which had a rank and a regiment. It is customary nowadays to

think of a square matrix, say with n rows and n columns. The

set of entries consists of n

2

symbols of the form Aα,thatis,one

Roman letter chosen from a list of n Roman letters, and one Greek

letter chosen from a list of n Greek letters, in such a way that the

entries are all different from one another. (If n is greater than 24,

ﬁnd lengthier alphabets.)

MATRICES 123

DEFINITION: A matrix of these n

2

entries satisfying the

condition that no row and no column contains the same Latin

or the same Greek letter twice is called a Graeco-Latin

Square.

For example,

⎡

⎢

⎢

⎢

⎣

Aα Bβ Cγ Dδ

Dγ Cδ Bα Aβ

Cβ Dα Aδ Bγ

Bδ Aγ Dβ Cα

⎤

⎥

⎥

⎥

⎦

is a 4-by-4 Graeco-Latin Square. It is easy to see that there is no

2-by-2 Graeco-Latin Square.

These squares are actually useful in experimental design, when

you want to test all possible combinations of two variables . By the

way, there is no 6-by-6 Graeco-Latin Square. Alas. (This fact was

proven by a very patient mathematician in 1901.)

In fact, after mathematicians showed that there is no 2-by-2

Graeco-Latin Square, and no 6-by-6 Graeco-Latin Square, it was

tempting to guess (as Euler did) that there was no 10-by-10 Graeco-

Latin Square. However, this turned out to be wrong (though you

are unlikely to ﬁnd one just by using trial-and-error without a

computer).

3

3

There are 8-by-8 Graeco-Latin Squares. In fact, Euler knew that there were n-by-n

Graeco-Latin Squares for any odd n or n divisible by 4. Euler conjectured that all 4k + 2-

by-4k + 2 cases (for k ≥ 0) were impossible, but in 1959 it was proven that all of them

are possible, except for the 2-by-2 and 6-by-6 cases!

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