152] 
475 
152. 
A MEMOIR ON THE THEORY OF MATRICES. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlviii. for 
the year, 1858, pp. 17—37. Received December 10, 1857,—Read January 14, 1858.] 
The term matrix might be used in a more general sense, but in the present 
memoir I consider only square and rectangular matrices, and the term matrix used 
without qualification is to be understood as meaning a square matrix; in this restricted 
sense, a set of quantities arranged in the form of a square, e.g. 
( a , h , c ) 
a', b', c' 
a", b", c" 
is said to be a matrix. The notion of such a matrix arises naturally from an 
abbreviated notation for a set of linear equations, viz. the equations 
X = ax + by + cz , 
Y = a'x + b'y 4- c'z, 
Z = a"x + b"y + c'z, 
may be more simply represented by 
(X, Y, Z) = ( a , b , c Jx, y, z), 
a', b', c' 
// 7 // _// 
CL , O j G 
and the consideration of such a system of equations leads to most of the fundamental 
notions in the theory of matrices. It will be seen that matrices (attending only to 
those of the same order) comport themselves as single quantities; they may be added, 
60—2