20 
ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
gives 
a ^ b , c 
)=( 
ma  ^ mb  ^ me 
a!,  b',  d 
md  ^ mb' , md 
a",  b\  d’ 
ma!',  mb",  me' 
as  the  rale  for  the  multiplication  of  a matrix  by  a single  quantity.  The  multiplier  m 
may  be  -written  either  before  or  after  the  matrix,  and  the  operation  is  therefore  com- 
mutative. We  have  it  is  clear  m(L+M)=mL+mM,  or  the  operation  is  distributive. 
9.  The  matrices  L and  rriL  may  be  said  to  be  similar  to  each  other ; in  particular,  if 
m=l,  they  are  equal,  and  if  m=  — 1,  they  are  opposite. 
10.  We  have,  in  particular. 
m(  1,  0,  0 ) = ( m,  0,  0 ), 
0,  1,  0 
o 
o 
0,  0,  1 
o' 
o" 
or  replacing  the  matrix  on  the  left-hand  side  by  unity,  we  may  write 
m=(  m,  0,  0 ), 
0,  m,  0 
0,  0,  m 
The  matrix  on  the  right-hand  side  is  said  to  be  the  single  quantity  m considered  as 
involving  the  matrix  rniity., 
11.  The  equations 
(X,  Y,  Z)=( 
a , b , c 
Xx,y,z),  {x,y,z)={ 
« , , y 
a!,  V , d 
i3',  y' 
a!',  b",  d' 
a",  y" 
give 
(X,  Y,  Z)  = ( A , B , C XI, Q=(  a,h,c\c.,^,y  II,  Q, 
A',  B',  C' 
A",  B",  C" 
and  thence,  substituting  for  the  matrix 
i3',  y' 
a",  j3",  y" 
( A,  B,  C ) 
A',  B',  C' 
A",  B",  C" 
its  value,  we  obtain 
( (a  , 5 , c Xa,  a',  a"),  (a  , 5 , c X/3,  /3',  /3"),  (a  , ^ , c Xr,  y',  y")  ) = ( a ,h  ,c  X a , , y 
(o',  S',  («',S',</Xy.Y,/) 
a! , b' , d 
(a",  S",  (a",  b\  c"X/3,  (3',  («",  S”,  c-'Xr,  y',  /) 
a!',  b",  d' 
a",  y" 
