140 THEORY OF COLLINEATIONS. 
lies in the 7th row and jth column is obtained by multiplying 
each element of the ith row of M by the corresponding ele- 
ment of 7th column of M, and adding the results. This law 
of composition is called Cayley’s rule. 
It is evident from the definition that the product MM, is 
not in general equal to the product M,M. Thus the product 
of two matrices is not definite unless the order of multiplica- 
tion be specified ; or we may say that the multiplication of 
matrices is not in general commutative, e. g., MM,+ M,M. 
Two matrices are said to be conjugate when the rows and 
columns of one are in the same order the columns and rows 
of the other. Thus 
ja bb a a a2 as|| 
M=\@ b& ec] and M’/=j\h & bs 
las bs cs Ci C2 C3 
are conjugate matrices. 
175. Determinant Form of the Resultant. Let T and T, 
be two collineations in homogeneous form as follows: 
Pa=uaxthy+az, Aw=Am+Ayt+naza , 
ips Pyi=a2u+ boy +c2z, ER: P1Yy2 = a+ Soyitjyen , Cay 
Pa =azsx+bsy +esz, Pi 22 = 4391+ 3yY1 +7321 - 
The collineation T, is obtained by eliminating «,, y,,z, from 
the above equations. This may be done as follows: Find the 
inverse of T by solving the three equations of T for x, y, z. 
Thus we get 
A 
— © = A+ Aoyit Asn , 
p 
A 
T-1 >) —y=Bin+ Boyit Bsa , (1) 
A 
— 2=C1%4+ Coyit+ Crz , 
where A is the determinant of T and A, B, ete., have the 
same meanings as in equations (3), Chapter II. 
