RESULTANTS. 139 
T, T,, T.,.. 7, in any order, is equal to the product of the 
determinants of the component collineations. Thus 
DENARAU OA... 
THEOREM 3. The determinant of the resultant of two or more 
collineations is equal to the product of the determinants of the com- 
ponent collineations. 
173. Definition and Notation of a Matrix. We shall now 
introduce the notion of a matrix and develop a few of its 
useful properties. A matrix may be defined as a system of 
mn quantities arranged in a rectangular array of m rows 
and n columns. We shall be concerned only with the case 
where m = n in which case we have a square matrix of order 
n. It is customary to distinguish a square matrix from a 
determinant by placing double bars on each side of the array. 
Thus : 
a hh oi} 
a2 be ce 
as bs ¢3 
M= 
is a square matrix of order 3. The determinant of the 
matrix is a thing distinct from the matrix itself. 
A matrix is said to be of rank if it contains at least one 
r-rowed determinant which does not vanish, while all deter- 
minants of an order higher than 7 which the matrix may con- 
tain are zero. If the determinant of the matrix does not 
vanish, the matrix is of rank n. 
The coefficients in the equations of a collineation form a 
matrix which may be taken as the analytic representation of 
the collineation. We may therefore speak of the matrix M 
of a collineation T and deal with the matrix instead of the 
collineation itself. 
174. Multiplication of two Matrices. The product of two 
square matrices is defined by the law of composition of two 
linear transformations as shown in Art. 172. This law of 
composition expressed in general terms gives the following 
definition: The product MM, of two matrices of the nth 
order is a matrix of the nth order in which the element that 
