36 
MB.  A.  CAYLEY  ON  THE  THEOBY  OF  MATBICES. 
and 
{ a d \ a!,  b',  c\  (^'))  = ( [a^  d\d^  e'\  (a,  d\h\  f),  («,  d'Jd,  g'),  {a,  d\d',  h')  ). 
h e\\e\f,  h'  ! i (b,  eXa\  e%  [b,  e^b',  /),  {b,  <;  Jc’,  y),  (},  e^d;.  h') 
f 1 (<=-  /!«'-  e'),  {c,  /IS',  /'),  (c,  /Ic-,  jr-),  (c,  /li'.  S') 
56.  In  the  particular  case  where  the  lines  and  columns  of  the  one  component  matrix 
are  respectively  equal  in  number  to  the  columns  and  lines  of  the  other  component 
matrix,  the  compound  matrix  is  square,  thus  we  have 
( a,  i,  c X ) = ( («.  b,  cja!,  b',  d),  (a,  b,  c^d',  e',  f)  ) 
\ d,  e,f 
and 
b',  e' 
d.  f 
( < d!  X a,  b,  c )=( 
b\  e' 
d,  e,  f 
(d,  e,  fja!,  b\  c'),  {d,  e,  f\d\  d,  f) 
(«',  d'Ja,  d),  («',  d'Xb,  e),  (a',  d'Jc,  f)  ) 
{b\  dXa,  d),  {V,  djb,  e%  (b',  dXc,  f)  | 
{d,  /!«,  d),  {d,  fXb,  e),  {d,  fjc,  f)  i 
The  two  matrices  in  the  case  last  considered  admit  of  composition  in  the  two  diiferent 
orders  of  arrangement,  but  as  the  resulting  square  matrices  are  not  of  the  same  order, 
the  notion  of  the  convertibility  of  two  matrices  does  not  apply  even  to  the  case  in 
question. 
57.  Since  a rectangular  matrix  cannot  be  compounded  with  itself,  the  notions  of 
the  inverse  or  reciprocal  matrix  and  of  the  powers  of  the  matrix  and  the  whole  resulting 
theory  of  the  functions  of  a matrix,  do  not  apply  to  rectangular  matrices. 
58.  The  notion  of  transposition  and  the  symbol  tr.  apply  to  rectangular  matrices,  the 
effect  of  a transposition  being  to  convert  a broad  matrix  into  a deep  one  and  reciprocally. 
It  may  be  noticed  that  the  symbol  tr.  may  be  used  for  the  purpose  of  expressing  the 
law  of  composition  of  square  or  rectangular  matrices.  Thus  treating  [a,  J,  c)  as  a 
rectangular  matrix,  or  representing  it  by  (a,  b,  c),  we  have 
tr.(  a',  b’,  d )=(  d ), 
' d 
d 
and  thence 
ff,  J,  c )tr.(^  d,  b',  d ^=z(a,  b,  cX.  b,  cX^^  b',  d), 
' ' ' b' 
d 
so  that  the  symbol 
(a,  5,  did,  b\  d) 
would  upon  principle  be  replaced  by 
(a,b,c^ tr.  ( d,  b',d  y. 
Ill  I 
