PHILLIPS AND MOORE. — LINEAR DISTANCE AND ANGLE. 



47 



which has a vaUie in the present discussion.] The matrix is zero when 

 all its elements are zero. In that case the points .4 and B have 

 proportional coordinates and hence coincide. If the matrix is not 

 zero it represents the line ^ J5 in the sense that from the matrix can 

 be obtained the coordinates of the line. Conversely if the line is 

 given a matrix can be formed by taking any two points on the line. 

 Different matrices representing the same line are multiples of any 

 one. For if A, B and P, Q are pairs of distinct points on the hne 



P =\A-\-\>B, 

 Q = fxi A -{- id2 B , 



and 



[PQ] 



Aj X-o 

 Ml M2 



[AB]. 



Thus a two-rowed matrix in addition to representing a line, has a 

 definite size. 



The matrix [A B] is in reality a set of six determinants 







taken in some definite order, 

 rowed matrix of six terms 



It can then be considered as a one- 



[AB]= II flj hk — cik hi\\. 



The sum of two matrices [A B] and [C B] is then a complex matrix, 

 each element of which is the sum of corresponding elements in [A B\ 

 and [C D]. In general this sum cannot be represented as a single 

 two-rowed matrix, just as the sum of corresponding Pliicker coordi- 

 nates of two lines are not in general coordinates of a line. For analy- 

 tical purposes we express this sum by simply writing the two matrices 

 with an addition sign between them. If, however, the lines A B and 

 C D intersect in a point P, we can find points Q and R on those lines 

 such that ^^ ^j ^ ^p ^^^ 



[C D] = [P R], 



Then 



[AB]^[CD] = [PQ]-^[PR]^ 



Pi 



Vk 



qk+n 



= [p{Q + m- 



For 



We can consider [.4 B] as a product of A and B. 



U(5 + C)]= [AB]+[AC] 



as we have just seen in the case of [P{Q -{- R)]. The process of multi- 

 plication consists in placing the second matrix under the matrix A 



