476 
A MEMOIR ON THE THEORY OF MATRICES. 
[152 
multiplied or compounded together, &c. : the law of the addition of matrices is pre 
cisely similar to that for the addition of ordinary algebraical quantities ; as regards 
their multiplication (or composition), there is the peculiarity that matrices are not in 
general convertible ; it is nevertheless possible to form the powers (positive or negative, 
integral or fractional) of a matrix, and thence to arrive at the notion of a rational 
and integral function, or generally of any algebraical function, of a matrix. I obtain 
the remarkable theorem that any matrix whatever satisfies an algebraical equation of 
its own order, the coefficient of the highest power being unity, and those of the 
other powers functions of the terms of the matrix, the last coefficient being in fact 
the determinant ; the rule for the formation of this equation may be stated in the 
following condensed form, which will be intelligible after a perusal of the memoir, 
viz. the determinant, formed out of the matrix diminished by the matrix considered 
as a single quantity involving the matrix unity, will be equal to zero. The theorem 
shows that every rational and integral function (or indeed every rational function) of 
a matrix may be considered as a rational and integral function, the degree of which 
is at most equal to that of the matrix, less unity; it even shows that in a sense, 
the same is true with respect to any algebraical function whatever of a matrix. One 
of the applications of the theorem is the finding of the general expression of the 
matrices which are convertible with a given matrix. The theory of rectangular 
matrices appears much less important than that of square matrices, and I have not 
entered into it further than by showing how some of the notions applicable to these 
may be extended to rectangular matrices. 
1. For conciseness, the matrices written down at full length will in general be 
of the order 3, but it is to be understood that the definitions, reasonings, and con 
clusions apply to matrices of any degree whatever. And when two or more matrices 
are spoken of in connexion with each other, it is always implied (unless the contrary 
is expressed) that the matrices are of the same order. 
2. The notation 
(a , b , c \x, y, z) 
a', V, c' 
x", b", c" 
represents the set of linear functions 
((a, b, c$x, y, z), (a, b', c'^x, y, z), (a", b", c"Jx, y, z)), 
so that calling these (X, F, Z), we have 
(X, Y, Z) = ( a , b , c \x, y, z) 
a', b', c' 
a", b", c" 
and, as remarked above, this formula leads to most of the fundamental notions in the 
theory.