| = | b, | c, | d | , | c, | a, | d | , | a, | b, | d | ||||||
| b′, | c′, | d′ | c′, | a′, | d′ | a′, | b′, | d′ | |||||||||
| b″, | c″, | d″ | c″, | a″, | d″ | a″, | b″, | d″ |
respectively.
6. Multiplication of two Determinants of the same Order.—The theorem is obtained very easily from the last preceding definition of a determinant. It is most simply expressed thus—
| (α, α′, α″), | (β, β′, β″), | (γ, γ′, γ″) | |||||||||||||||||||
| (a, | b, | c | ) | " | " | " | = | a, | b, | c | . | α, | β, | γ | , | ||||||
| (a′, | b′, | c′ | ) | " | " | " | a′, | b′, | c′ | α′, | β′, | γ′ | |||||||||
| (a″, | b″, | c″ | ) | " | " | " | a″, | b″, | c″ | α″, | β″, | γ″ |
where the expression on the left side stands for a determinant, the terms of the first line being (a, b, c)(α, α′, α″), that is, aα + bα′ + cα″, (a, b, c)(β, β′, β″), that is, aβ + bβ′ + cβ″, (a, b, c)(γ, γ′, γ″), that is aγ + bγ′ + cγ″; and similarly the terms in the second and third lines are the life functions with (a′, b′, c′) and (a″, b″, c″) respectively.
There is an apparently arbitrary transposition of lines and columns; the result would hold good if on the left-hand side we had written (α, β, γ), (α′, β′, γ′), (α″, β″, γ″), or what is the same thing, if on the right-hand side we had transposed the second determinant; and either of these changes would, it might be thought, increase the elegance of the form, but, for a reason which need not be explained,[[2]] the form actually adopted is the preferable one.
To indicate the method of proof, observe that the determinant on the left-hand side, qua linear function of its columns, may be broken up into a sum of (3³ =) 27 determinants, each of which is either of some such form as
| = αβγ′ | a, | a, | b | , | ||
| a′, | a′, | b′ | ||||
| a″, | a″, | b″ |
where the term αβγ' is not a term of the αβγ-determinant, and its coefficient (as a determinant with two identical columns) vanishes; or else it is of a form such as
| = αβ′γ″ | a, | b, | c | , | ||
| a′, | b′, | c′ | ||||
| a″, | b″, | c″ |