BOUNDARY PROBLEMS AND DEVELOPMENTS. 53 



That it is in general not commutative is a consequence of the fact that 



n n 



2 dikbkj is in general not equal to 2 bikCikj. The rearrangement of 



the factors in a matrix product is, therefore, as a rule, not permissible. 



The multiplication of a scalar into a matrix has simply the effect of 

 multiplying each element of the matrix by the scalar. Thus if k is a 

 scalar, then k A = Ak = (A;a t -,-). Conversely, any factor common to 

 all the elements of a matrix can be factored from the matrix. 



Two special matrices must be mentioned, namely = (0), the 

 zero matrix, and / = (5;;), the unit matrix, where 5 t y = when i 4=j, 

 8u = 1 . These matrices satisfy respectively the relations 



AO = OA = 

 and AI = IA = A. 



The determinant formed from the elements of a matrix without 

 changing the order of the array is called the determinant of the matrix. 

 The alternative notations | an | or | A | for the determinant of the 

 matrix (a»,-) will be used. 



Given a matrix A it follows that if | A \ ^z then there exists a 

 unique solution in the x's for each of the linear systems 



n 



2 dik Xkj = 8ij , i = 1, 2, . . .n, 



where jo ranges from 1 to n. This means that there exists a unique 

 set of n 2 quantities xa such that 



n 



2 a ik x k j = Sij, i, j, = 1, 2, ... n, 



i.e. there exists a unique matrix (a* i; ) such that 



A-(x i3 ) = I. 



This matrix (xi } ) is denoted by the symbol A -1 and is called the inverse 

 of A. From its derivation it is seen to satisfy the relation AA~ X = I. 

 Either of the relations, AX = or XA = 0, leads, on the assump- 

 tion that | ^4 | =4= 0, to the conclusion X = 0, as is evident from the 

 theory of the systems of linear equations to which the matrix equations 



