original determinant is＝0, and therefore the determinant itself is＝0; that is, the linear equations give
| x | a, | b, | c | − | d, | b, | c | ＝0; | ||||
| a′, | b′, | c′ | d′, | b′, | c′ | |||||||
| a″, | b″, | c″ | d″, | b″, | c″ |
which is the result obtained above.
We might in a similar way find the values of y and z, but there is a more symmetrical process. Join to the original equations the new equation
αx + βy + γz ＝ δ;
a like process shows that, the equations being satisfied, we have
| α, | β, | γ, | δ | ＝0; | ||
| a, | b, | c, | d | |||
| a′, | b′, | c′, | d′ | |||
| a″, | b″, | c″, | d″ |
or, as this may be written,
| α, | β, | γ, | − δ | a, | b, | c | ＝0; | |||||
| a, | b, | c, | d | a′, | b′, | c′ | ||||||
| a′, | b′, | c′, | d′ | a″, | b″, | c″ | ||||||
| a″, | b″, | c″, | d″ |
which, considering δ as standing herein for its value αx + βy + γz, is a consequence of the original equations only: we have thus an expression for αx + βy + γz, an arbitrary linear function of the unknown quantities x, y, z; and by comparing the coefficients of α, β, γ on the two sides respectively, we have the values of x, y, z; in fact, these quantities, each multiplied by
| a, | b, | c | , | ||
| a′, | b′, | c′ | |||
| a″, | b″, | c″ |
are in the first instance obtained in the forms
| 1 | , | 1 | , | 1 | ; | |||||||||||||||
| a, | b, | c, | d | a, | b, | c, | d | a, | b, | c, | d | |||||||||
| a′, | b′, | c′, | d′ | a′, | b′, | c′, | d′ | a′, | b′, | c′, | d′ | |||||||||
| a″, | b″, | c″, | d″ | a″, | b″, | c″, | d″ | a″, | b″, | c″, | d″ |
but these are
| ＝ | b, | c, | d | , − | c, | d, | a | , | d, | a, | b | , | ||||||
| b′, | c′, | d′ | c′, | d′, | a′ | d′, | a′, | b′ | ||||||||||
| b″, | c″, | d″ | c″, | d″, | a″ | d″, | a″, | b″ |
or, what is the same thing,
| ＝ | 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,[1] 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″ |
that is, every term which does not vanish contains as a factor the abc-determinant last written down; the sum of all other factors ± αβ′γ″ is the αβγ-determinant of the formula; and the final result then is, that the determinant on the left-hand side is equal to the product on the right-hand side of the formula.
7. Decomposition of a Determinant into complementary Determinants.—Consider, for simplicity, a determinant of the fifth order, 5＝2 + 3, and let the top two lines be
| a, | b, | c, | d, | e |
| a′, | b′, | c′, | d′, | e′ |
then, if we consider how these elements enter into the determinant, it is at once seen that they enter only through the determinants of the second order
| a, | b | , | ||
| a′, | b′ |
&c., which can be formed by selecting any two columns at pleasure. Moreover, representing the remaining three lines by
| a″, | b″, | c″, | d″, | e″ |
| a‴, | b‴, | c‴, | d‴, | e‴ |
| a⁗, | b⁗, | c⁗, | d⁗, | e⁗ |
it is further seen that the factor which multiplies the determinant formed with any two columns of the first set is the determinant of the third order formed with the complementary three columns of the second set; and it thus appears that the determinant of the fifth order is a sum of all the products of the form
| a, | b | c″, | d″, | e″ | , | |||||
| a′, | b″ | c‴, | d‴, | e‴ | ||||||
| c⁗, | d⁗, | e⁗ |
the sign ± being in each case such that the sign of the term ± ab′c″d‴e⁗ obtained from the diagonal elements of the component determinants may be the actual sign of this term in the determinant of the fifth order; for the product written down the sign is obviously +.
Observe that for a determinant of the n-th order, taking the decomposition to be 1 + (n − 1), we fall back upon the equations given at the commencement, in order to show the genesis of a determinant.
8. Any determinant
| a, | b | ||
| a′, | b′ |
formed out of the elements of the original determinant, by selecting the lines and columns at pleasure, is termed a minor of the original determinant; and when the number of lines and columns, or order of the determinant, is n−1, then such determinant is called a first minor; the number of the first minors is＝n 2, the first minors, in fact, corresponding to the several elements of the determinant—that is, the coefficient therein of any term whatever is the corresponding first minor. The first minors, each divided by the determinant itself, form a system of elements inverse to the elements of the determinant.
A determinant is symmetrical when every two elements symmetrically situated in regard to the dexter diagonal are equal to each other; if they are equal and opposite (that is, if the sum of the two elements be＝0), this relation not extending to the diagonal elements themselves, which remain arbitrary, then the determinant is skew; but if the relation does extend to the diagonal terms (that is, if these are each＝0), then the determinant is skew symmetrical; thus the determinants
| a, | h, | g | ; | a, | ν, | −μ | ; | 0, | ν, | −μ | ||||||
| h, | b, | f | −ν, | b, | λ | −ν, | 0, | λ | ||||||||
| g, | f, | c | μ, | −λ, | c | μ, | −λ, | 0 |
are respectively symmetrical, skew and skew symmetrical.
- ↑ The reason is the connexion with the corresponding theorem for the multiplication of two matrices.
