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-- ([alpha], [alpha]', [alpha]"), ([beta],[beta]',[beta]"), ([gamma],[gamma]',[gamma]") +---------------------------------------+ (a , b , c )| " " " | = (a', b', c')| " " " | (a", b", c")| " " " | = |a , b , c |. |[alpha] , [beta] , [gamma] |, |a', b', c'| |[alpha]', [beta]', [gamma]'| |a", b", c"| |[alpha]", [beta]", [gamma]"| where the expression on the left side stands for a determinant, the terms of the first line being (a, b, c)([alpha], [alpha]', [alpha]"), that is, a[alpha] + b[alpha]' + c[alpha]", (a, b, c)([beta], [beta]', [beta]"), that is, a[beta] + b[beta]' + c[beta]", (a, b, c)([gamma], [gamma]', [gamma]"), that is a[gamma] + b[gamma]' + c[gamma]"; 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 ([alpha], [beta], [gamma]), ([alpha]', [beta]', [gamma]'), ([alpha]", [beta]", [gamma]"), 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 = [alpha][beta][gamma]'|a , a , b |, |a', a', b'| |a", a", b"| where the term [alpha][beta][gamma]' is not a term of the [alpha][beta][gamma]-determinant, and its coefficient (as a determinant with two identical columns) vanishes; or else it is of a form such as = [alpha][beta]'[gamma]"|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 ± [alpha][beta]'[gamma]" is the [alpha][beta][gamma]-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.