ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
19 
required,  say,  the  matrix  zero,  or  (as  the  case  may  be)  the  matrix  unity  of  such  au  order. 
The  matrix  zero  may  for  the  most  part  be  represented  simply  by  0,  and  the  matrix 
unity  by  1. 
4.  The  equations 
(X,  Y.  Z)  = ( 
a , b , c 
(X',Y',  Z')=( 
a , 3 , 7 
a',  b' , d 
f3',  y' 
a",  b",  d' 
3",  y" 
give 
(X+X',  Y+T,  Z+Z')=(  « ^ +/3  , +y  y.  Z) 
a! -{-cd , c' 
j»4-/3",  c"+/ 
and  this  leads  to 
( u ~\~(z  , b ^ ~\~y 
)=( 
a , b , c 
) + ( 
0^  ^ f3  , 7 
1 a' -fa',  b'+(3',  c'-fy' 
a\  b',  d 
3',  r' 
: a"+a",  b"+(i",  d'+f 
a",  b",  c" 
3",  7" 
as  the  rule  for  the  addition  of  matrices ; that  for  their  subtraction  is  of  course  similar 
to  it. 
5.  A matrix  is  not  altered  by  the  addition  or  subtraction  of  the  matrix  zero,  that  is, 
we  have  M+0=M. 
The  equation  L=M,  which  expresses  that  the  matrices  L,  M are  equal,  may  also  be 
written  in  the  form  L — M=0,  i.  e.  the  difference  of  two  equal  matrices  is  the  matrix 
zero. 
6.  The  equation  L= — M,  written  in  the  form  L-|-M=0,  expresses  that  the  sum  of 
the  matrices  L,  M is  equal  to  the  matrix  zero,  the  matrices  so  related  are  said  to  be 
opposite  to  each  other ; in  other  words,  a matrix  the  terms  of  which  are  equal  but 
opposite  in  sign  to  the  terms  of  a given  matrix,  is  said  to  be  opposite  to  the  given 
matrix. 
7.  It  is  clear  that  we  have  L-f-M=M+L,  that  is,  the  operation  of  addition  is  com- 
mutative, and  moreover  that  (L4-M)4-N=L+(M+N)  = L+M+N,  that  is,  the  opera- 
tion of  addition  is  also  associative. 
8.  The  equation 
(X,  Y,  Z)=(  a , ^ , 6*  frux.,  my,  mz) 
a',  b' , d 
a",  h\  d' 
written  under  the  forms 
(X,  Y,  Z)=w(  a , b , c fx,  y,  2)=(  ma  , mb  , me  fx,  y,  z) 
a',  b' , d 
ma! , mb' , md 
a",  b\  d’ 
D 2 
ma!',  mb",  md' 
