ax + by + cz = d,
a'x + b'y + c'z = d',
a"x + b"y + c"z = d",
and proceeding to solve them by the so-called method of cross
multiplication, we multiply the equations by factors selected in such a
manner that upon adding the results the whole coefficient of y becomes =
0, and the whole coefficient of z becomes = 0; the factors in question
are b'c" - b"c', b"c - bc", bc' - b'c (values which, as at once seen,
have the desired property); we thus obtain an equation which contains on
the left-hand side only a multiple of x, and on the right-hand side a
constant term; the coefficient of x has the value
a(b'c" - b"c') + a'(b"c - bc") + a"(bc' - b'c),
and this function, represented in the form
|a, b, c |,
|a', b', c'|
|a", b", c"|
is said to be a determinant; or, the number of elements being 3², it is
called a determinant of the third order. It is to be noticed that the
resulting equation is
|a, b, c | x = |d, b, c |
|a', b', c'| |d', b', c'|
|a", b", c"| |d", b", c"|
where the expression on the right-hand side is the like function with d,
d', d" in place of a, a', a" respectively, and is of course also a
determinant. Moreover, the functions b'c" - b"c', b"c - bc", bc' - b'c
used in the process are themselves the determinants of the second order
|b', c'|, |b", c"|, |b, c |.
|b", c"| |b, c | |b', c'|
We have herein the suggestion of the rule for the derivation of the
determinants of the orders 1, 2, 3, 4, &c., each from the preceding one,
viz. we have
|a| = a,
|a, b | = a|b'| - a'|b|.
|a', b'|
|a, b, c | = a|b', c'| + a'|b", c"| + a"|b, c |,
|a', b', c'| |b", c"| |b , c | |b', c'|
|a", b", c"|
|a, b , c , d | = a|b', c', d' | - a'|b" , c" , d" | +
|a', b' , c' , d' | |b", c", d" | |b"', c"', d"'|
|a", b" , c" , d" | |b"', c"', d"'| |b , c , d |
|a"', b"', c"', d"'|
+ a"|b"', c"', d"'| - a"'|b , c, d |,
|b , c , d | |b', c', d'|
|b' , c' , d' | |b", c", d"|
and so on, the terms being all + for a determinant of an odd order, but
alternately + and - for a determinant of an even order.