24 
MR.  A.  CAYLEY  ON  THE  THEORY  OE  ]yLiTRICES. 
quantity  involving  the  matrix  unity.  And  this  is  the  general  theorem,  -viz.  the  deter- 
minant, having  for  its  matrix  a given  matrix  less  the  same  matrix  considered  as  a single 
quantity  involving  the  matrix  unity,  is  equal  to  zero. 
22.  The  following  symbolical  representation  of  the  theorem  is,  I think,  worth  no- 
ticing : let  the  matrix  M,  considered  as  a single  quantity,  be  represented  by  ]SI,  then 
mating  1 to  denote  the  matrix  unity,  M.l  will  represent  the  matrix  M,  considered  as  a 
single  quantity  involving  the  matrix  unity.  Upon  the  like  principles  of  notation,  l.M 
will  represent,  or  may  be  considered  as  representing,  simply  the  matrix  M,  and  the 
theorem  is 
Det.  (T.M-M.1)  = 0. 
23.  I have  verified  the  theorem,  in  the  next  simplest  case,  of  a matrix  of  the  order  3, 
viz.  if  M be  such  a matrix,  suppose 
M=(  a,  h,  c ), 
d,  e,  f 
g,  h,  i 
then  the  derived  determinant  vanishes,  or  we  have 
a — M,  b 
, c 
= 0, 
d , e-M,  / 
g , A , i— M 
or  expanding, 
M^—{a-{-e-\-i)M,^-\-{ei-\-ia-^ae—fh—cg—bd)'M.—{aei-{-bfg-\-cdh—afJi—bdi—ceg)—0 ; 
but  I have  not  thought  it  necessary  to  undertake  the  labour  of  a formal  proof  of  the 
theorem  in  the  general  case  of  a matrix  of  any  degree. 
24.  If  we  attend  only  to  the  general  form  of  the  result,  we  see  that  any  matrix  what- 
ever satisfies  an  algebraical  equation  of  its  own  order,  which  is  in  many  cases  the  mate- 
rial part  of  the  theorem. 
25.  It  follows  at  once  that  every  rational  and  integral  function,  or  indeed  ever}' 
rational  function  of  a matrix,  can  be  expressed  as  a rational  and  integral  function  of  an 
order  at  most  equal  to  that  of  the  matrix,  less  unity.  But  it  is  important  to  consider 
how  far  or  in  what  sense  the  like  theorem  is  true  with  respect  to  irrational  functions  of 
a matrix.  If  we  had  only  the  equation  satisfied  by  the  matrix  itself,  such  extension 
could  not  be  made ; but  we  have  besides  the  equation  of  the  same  order  satisfied  by  the 
irrational  function  of  the  matrix,  and  by  means  of  these  two  equations,  and  the  equa- 
tion by  which  the  irrational  function  of  the  matrix  is  determined,  we  may  express  the 
irrational  function  as  a rational  and  integral  function  of  the  matrix,  of  an  order  equal  at 
most  to  that  of  the  matrix,  less  unity;  such  expression  will  however  in'\olve  the  coeffi- 
cients of  the  eguation  satisfied  by  the  irrational  function  which  are  functions  (m  number 
equal  to  the  order  of  the  matrix)  of  the  coefficients  assumed  unknown,  of  the  irrational 
function  itself.  The  transformation  is  nevertheless  an  important  one,  as  reducing  the 
