ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
19.  The  notion  of  the  inverse  or  reciprocal  matrix  once  established,  the  other  nega- 
tive integer  powers  of  the  original  matrix  are  positive  integer  powers  of  the  inverse  or 
reciprocal  matrix,  and  the  theory  of  such  negative  integer  powers  may  be  taken  to  be 
known.  The  further  discussion  of  the  fractional  powers  of  a matrix  will  be  resumed  in 
the  sequel. 
20.  The  positive  integer  power  L™  of  the  matrix  L may  of  course  be  multiplied  by 
any  matrix  of  the  same  degree,  such  multipher,  however,  is  not  in  general  convertible 
with  L ; and  to  preserve  as  far  as  possible  the  analogy  with  ordinary  algebraical  func- 
tions, we  may  restrict  the  attention  to  the  case  where  the  multiplier  is  a single  quan- 
tity, and  such  convertibility  consequently  exists.  We  have  in  this  manner  a matrix  cL"‘, 
and  by  the  addition  of  any  number  of  such  terms  we  obtain  a rational  and  integral  func- 
tion of  the  matrix  L. 
21-.  The  general  theorem  before  refen-ed  to  will  be  best  understood  by  a complete 
development  of  a particular  case.  Imagine  a matrix 
M=(  «,  1)  ), 
1 c,  (Z  j 
and  form  the  determinant 
« — M,  h , 
' c,  d—M 
the  developed  expression  of  this  determinant  is 
(a+ fZ)M‘ -f  ; 
the  values  of  M'*,  M',  M®  are 
( a^+bc  , b{a-^d)  ),  ( a,  5 ),  ( 1,  0 ), 
1 c{a-\-d),  d^-\-bc  | \ c,  d \ | 0,  1 | 
and  substituting  these  values  the  determinant  becomes  equal  to  the  matrix  zero,  viz.  we 
have 
a— M,  b =(  (d-\-bc  , b{a-\-d)  ) — {a-^d)  (i,  b )-\-(ad—bc)  ( 1,  0 ) 
c , ri-M 
c{a-^d),  d^-\-bc 
c,  d 
I 0,  1 I 
= ( {a^-^bc)  — {a-\-d)a-{-{ad—bc),  b{a-\-d)  — {a-\-d)b  ) = ( 0 ), 
I c(a-\-d)—(a-{-d)c  , d^-\-bc—{a-\-d)d-\-ad—bG  | | 0,  0 | 
that  is, 
a — M,  b =0 
c , M 
where  the  matrix  of  the  determinant  is 
( a,  i )-M(  1,  0 ), 
c,  d 
0,  1 
that  is,  it  is  the  original  matrix,  diminished  by  the  same  matrix  considered  as  a single 
