PHILLIPS AND MOORE. — LINEL\R DISTANCE AND ANGLE. 



53 



point or hyperplane coordinates. If the matrices are of different 

 kinds we replace one of them by its contragredient form in such a way 

 that the new matrices have a sum of rows equal to or less than n. 

 Thus in every case of a product there is a definite result that has a 

 meaning. We call this the product of those factors. 



7. Reduction formulae. We have just found that in expressing 

 the product of two matrices when the sum of the rows is greater than 

 n, we must change to contragredient forms. We shall now derive 

 certain reduction formulae by which we obtain the same results 



For this purpose let 



without that change. 



[Ai A2. . . .Ap] = [ap+i_ ap+2- ■ ■ -cij 

 [B, B2....B,] = [/3,^i, ^,^2..../5j. 



We shall now prove that in the determinant 



an 



ai2 



. . . ain 



■an 



■■ai2 



'In 



^n. 



^n. 



^n,n 13. 



i8«.2 



Pn, n 



each minor from the n — p rows of as and w — q rows of /S's is equal to 

 its algebraic compliment. To prove this we first show that if such a 

 minor M contains a product of a minor .1 whose order is ?i —p in the 

 as by a minor B of order n — q in the |S's, then the algebraic comple- 

 ment of M contains a product of minors respectively equal to A 

 and B. Since A is contained in the principal minor | On • • • a^nl, 

 if B is in its complement \ bu. . . ./3,„,|, the result is obvious. For 

 the algebraic complements of A and B respectively in those prin- 

 cipal minors are the terms required. If B is not contained entirely 

 in the principal minor, there is a minor B', in | ^n . . . ./3„„ | containing 

 the same letters as B and in the same order (but having perhaps 

 different signs). In the algebraic complement of a minor 31' contain- 

 ing A, B' is then a term C D = A B' where C is a minor of p rows of 

 a's and D a minor of q rows of 6's. If now we permute the columns 



