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3.4 WhyDo It This Way 93
3.4 WHY DO IT THIS WAY

If you were given the task of formulating a definition for composing two ma-
trices A and B in some sort of “natural” multiplicative fashion, your first
attempt would probably be to compose A and B by multiplying correspond-
ing entries—much the same way matrix addition is defined. Asked then to defend
the usefulness of such a definition, you might be hard pressed to provide a truly
satisfying response. Unless a person is in the right frame of mind, the issue of
deciding how to best define matrix multiplication is not at all transparent, es-
pecially if it is insisted that the definition be both “natural” and “useful.” The
world had to wait for Arthur Cayley to come to this proper frame of mind.
As mentioned in §3.1, matrix algebra appeared late in the game. Manipula-
tion on arrays and the theory of determinants existed long before Cayley and his
theory of matrices. Perhaps this can be attributed to the fact that the “correct”
way to multiply two matrices eluded discovery for such a long time.
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Around 1855, Cayley became interested in composing linear functions. In
particular, he was investigating linear functions of the type discussed in Example
3.3.2. Typical examples of two such functions are

x 1 ax 1 + bx 2 x 1 Ax 1 + Bx 2
f(x)= f = and g(x)= g = .
x 2 cx 1 + dx 2 x 2 Cx 1 + Dx 2
Consider, as Cayley did, composing f and g to create another linear function

Ax 1 + Bx 2 (aA + bC)x 1 +(aB + bD)x 2
h(x)= f g(x) = f = .
Cx 1 + Dx 2 (cA + dC)x 1 +(cB + dD)x 2
It was Cayley’s idea to use matrices of coefficients to represent these linear
functions. That is, f, g, and h are represented by

a b A B aA + bC aB + bD
F = , G = , and H = .
c d C D cA + dC cB + dD
After making this association, it was only natural for Cayley to call H the
composition (or product)of F and G, and to write

a b A B aA + bC aB + bD
= . (3.4.1)
c d C D cA + dC cB + dD
In other words, the product of two matrices represents the composition of the
two associated linear functions. By means of this observation, Cayley brought to
life the subjects of matrix analysis and linear algebra.
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Cayley was not the first to compose linear functions. In fact,Gauss used these compositions
as early as 1801,but not in the form of an array of coefficients. Cayley was the first to make
the connection between composition of linear functions and the composition of the associated
matrices. Cayley’s work from 1855 to 1857 is regarded as being the birth of our subject.