100 



cator, the other allowed the currents of the thermo-elements to pass 

 either so as to strengthen, or so as to oppose each other. 



The foregoing experiments were carried out in the Physical Ca- 

 binet at Heidelberg, under the direction of Professor Kirchhoff, to 

 whose advice and assistance I am much indebted. 



III. " A Memoir on the Theory of Matrices." By ARTHUR 

 CAYLEY, Esq., F.R.S. Received December 10, 1857. 



[Abstract.] 



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 quan- 

 tities arranged in the form of a square, e. g. 

 ( a, b, c ) 

 a', b r , c' 

 a", 6", c" 



is said to be a matrix. The notation of such a matrix arises naturally 

 from an abbreviated notation for a set of linear equations, viz. the 

 equations 



may be more simply represented by 



(X, Y, Z)=( a, b, c Jar, y, z) 



a', b', c' 



a",b",c" 



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 degree) com- 

 port themselves as single quantities ; they may be added, multiplied, 

 or compounded together, &c. : the law of the addition of matrices is 

 precisely similar to that for the addition of ordinary algebraical quan- 

 tities ; as regards their multiplication (or composition), there is the 

 peculiarity that matrices are not in general convertible ; it is never- 

 theless possible to form the powers (positive or negative, integral or 



