48 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



to form a two rowed matrix [A B]. We shall call this the yrogresshe 

 product. From the definition it is evident that 



{AB] = - [B A] 

 and [A A] = 0. 



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

 plane ABC are proportional to the three rowed determinants in 

 the matrix 



Oi 02 03 04 



[ABC]= 6i 62 &3 64 



Ici C2 Cz d 



This matrix represents the plane in the sense that from it we can 

 determine the coordinates of the plane. Conversely to represent any 

 plane as a matrix we take three non-coUinear points of the plane and 

 form the matrix from them. The elements of such a matrix are the 

 three-rowed determinants belonging to it. In reality we are consider- 

 ing this as a one-rowed matrix of four terms (equal to the three-rowed 

 determinants in [A B C]) arranged in some definite order. Two 

 matrices of this kind will be called equal if corresponding elements 

 are equal and are added by adding corresponding elements. 



If P, Q, R are any three points of the plane determined hy A, B, C, 



P=\A-\-\2B-{- XgC, 

 Q = ^i,A + ^i.B + utsC, 

 R= p^A-\- V2B-{- v^C. 



Consequently 



[p Q m 



[A B C]. 



Thus a matrix [P Q R] in addition to representing a plane has a 

 definite size. The vanisliing of [P Q R] signifies that P, Q, R lie 

 on a line. 



The matrix [A B C] can be regarded as a product of [A B] and C, 

 A and [B C] or oi A, B and C, the process of multiplication consisting 

 always of placing the first matrix at the top and the others in order 

 under it to form a single matrix.^ 



2 In this multiplication each matrix must have four columns. If instead of 

 [A B] we have a complex the operation must be performed distributively on 

 each two-rowed matrix of the sum. For purposes of addition we regard our 

 quantities as matrices of one row but for purposes of multipHcation as matrices 

 or sums of matrices of four columns. 



