ME.  A.  CAYLEY  ON  THE  THEOEY  OE  MATEICES. 
31 
36.  Two  matrices  such  as 
i a,  b ), 
I c,  d 
( a,  c ), 
b,  d 
are  said  to  be  formed  one  from  the  other  by  transposition,  and  this  may  be  denoted  by 
the  symbol  tr. ; thus  we  may  write 
( a,  c ) = tr.( 
\ b,  d\  I 
a,  b ), 
c,  d 
The  effect  of  two  successive  transpositions  is  of  course  to  reproduce  the  original  matrix. 
37.  It  is  easy  to  see  that  if  M be  any  matrix,  then 
and  in  particular. 
(tr.  M7=tr.  (M^), 
(tr.  M)“^=  tr.  (M"*). 
38.  If  L,  M be  any  two  matrices, 
tr.  (LM)=  tr.  M.  tr.  L, 
and  similarly  for  three  or  more  matrices,  L,  M,  N,  &c., 
tr.  (LMN)=  tr.  N.  tr.  M.  tr.  L,  &c. 
40.  A matrix  such  as 
( g ) 
h,  b,f 
g^f^  c 
which  is  not  altered  by  transposition,  is  said  to  be  symmetrical. 
41,  A matrix  such  as 
( 0,  ^ ) 
— v,  0,  'k 
p,  —X,  0 
which  by  transposition  is  changed  into  its  opposite,  is  said  to  be  skew  symmetrical. 
42.  It  is  easy  to  see  that  any  matrix  whatever  may  be  expressed  as  the  sum  of  a sym- 
metrical matrix,  and  a skew  symmetrical  matrix ; thus  the  form 
( « , g—ijj  ) 
h—v-,  ^ ■> 
which  may  obviously  represent  any  matrix  whatever  of  the  order  3,  is  the  sum  of  the 
two  matrices  last  before  mentioned. 
43.  The  following  formulse,  although  little  more  than  examples  of  the  composition  of 
transposed  matrices,  may  be  noticed,  viz. 
(a,  c)  = ( , ac-\-bd  } 
ac-\-bd,  c^-\-d'^  j 
