392 



MOORE AND PHILLIPS. 



that is, if the points A and B coincide. One matrix is a multiple of 

 another if corresponding elements are proportional. If P, Q are any 

 two distinct points on the line AB, [PQ] is a multiple of [AB], For 

 numbers Xi, Xo, mi. M2 can then be found such that 



P = X1.4 + XoB, 



Q = lJiiA-\- fxoB. 

 From these equations, by use of (2) it is easy to show that 



Xi X2 



[PQ] = 



M1M2 



AB]. 



Thus a matrix in addition to reprcserUing a line has a definite magnitude. 

 The sum of two matrices [AB] and [CD] is the matrix each element 

 of which is the sum of corresponding elements of [AB] and [CD]. 

 In general the elements of this sum are not the two-rowed determi- 

 nants of a matrix of the type (3), just as the sum of corresponding 

 Pliicker coordinates of two lines are not in general coordinates of a 

 line. A matrix 



C12 Cl3 CU C03 C24 ^34 II = II Cifc 



(5) 



whose elements Cik are the determinants of (3) will be called simple. 

 Such a matrix represents a line. If, however, the elements c^ cannot 

 be so represented, the matrix will be called complex, we shall say that 

 this represents a comple.r lifie. The relation of this to the linear com- 

 plex will be shown later (§5). In what follows the word line will 

 always mean simple line. 



If the lines AB and CD intersect in a point P, we can find points 

 Q and R on those lines and assign to them such magnitudes that 



Then 



[AB] = [PQ], [CD] = [PR]. 



[AB] + [CD] = [PQ] + [PR] 

 vMk + n) - piMi +r,)\\ = [P{Q + R)]. 



(6) 



Therefore the sum of two intersecting lines is a line. 



If the points A, B, C are not collinear, the coordinates of the plane 

 ABC are the three-rowed determinants of the matrix. 



(7) 



