ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
37 
it  is  however  more  convenient  to  retain  the  symbol 
(a,  b,  c^a',  b',  c'). 
Hence  introducing  the  symbol  tr.  only  on  the  left-hand  sides,  we  have 
a,  b,  c )tr.  ( b\  d ) = ((«,  b,  cja',  b\  d),  {a,  b,  c^d',  d,  f)  ), 
\d,e,f\  i a,  «',  / 1 I (d,  e,  /!«',  b',  </),  (d,  e,  fj^d',  d,  f)  \ 
or  to  take  an  example  involving  square  matrices, 
( «,  ^ )tr.  ( a',  ^'  ) = ( «')  ). 
\ d,  e \ I d!,  d | j {d,  eja',  b'),  (d,  e^d',  d)  i 
so  that  in  the  composition  of  matrices  (square  or  rectangular),  when  the  second  or 
nearer  component  matrix  is  expressed  as  a matrix  preceded  by  the  symbol  tr.,  any  line 
of  the  compound  matrix  is  obtained  by  compounding  the  corresponding  line  of  the  first 
or  further  component  matrix  successively  with  the  several  lines  of  the  matrix  whicli 
preceded  by  tr.  gives  the  second  or  nearer  component  matrix.  It  is  clear  that  the  terms 
‘ symmetrical’  and  ‘ skew  symmetrical’  do  not  apply  to  rectangular  matrices. 
