708 X. ON THE STRUCTURE OF 'MATTER'
are together equivalent to one single matrix equation. There are several ways in which the notation can be simplified.
The difference between a determinant and a matrix is subtle, but important. By a determinant we understand, by definition, a certain homogeneous polynomial of the n-th degree, in the n2 elements a,,-. Accordingly, a determinant gives a definite number when calculated.
But in many instances we are interested in the table, or the n2 elements arranged in a certain order but not combined into a polynomial. Such an array, or table, is called a matrix. Thus, from this point of view, a matrix does not represent a definite quantity, but a system of quantities, and so a matrix is not a determinant.
We can illustrate this difference by an example. If we take a determinant of the second order
and change the rows into columns, or vice versa, thus:
the value of both determinants will be equal; namely,
by the definition rule already given; yet the matrices of the two determinants are different.
Although different, a determinant nevertheless defines a matrix, called the matrix of the determinant; conversely, a matrix defines a determinant, called the determinant of the matrix.
We have said that a matrix does not represent a quantity, while a determinant does. At this stage, and from this point of view, we may say so legitimately. However, we might eventually treat a matrix as a quantity also; but for this purpose we should have to enlarge the meaning of the term 'quantity'.
In our present use of the term 'quantity' we mean the real and complex quantities of ordinary algebra.
It may be said that mathematicians have had a peculiar tendency, which has proven of great value in the development of mathematics, gradually to extend the meaning of terms in order to embrace new notions as they arise. For instance, we have enlarged the primitive meaning applied to positive integers to embrace negative numbers, which formerly would not have been considered as quantities. Similarly, if we here use the ordinary notion of algebraic quantity, then a matrix is not a quantity but a system of quantities. The problem is, how shall we enlarge this meaning to include the matrices?