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ME.  A.  CAYLEY  OY  THE  THEOEY  OF  MATEICES. 
in  the  other  case,  where  the  assumed  equation  is  of  an  order  inferior  to  that  of  the 
matrix,  the  condition  is  necessary,  but  it  is  not  sufficient. 
30.  The  equation  satisfied  by  the  matrix  may  be  of  the  form  M"=l ; the  matrix  is 
in  this  case  said  to  be  periodic  of  the  ?^th  order.  The  preceding  considerations  apply  to 
the  theory  of  periodic  matrices ; thus,  for  instance,  suppose  it  is  required  to  find  a matiix 
of  the  order  2,  which  is  periodic  of  the  second  order.  Writing 
M = ( a,  h ), 
I I 
\ c,  d \ 
we  have 
and  the  assumed  equation  is 
'fhese  equations  will  be  identical  if 
a-\-d=^,  ad — hc=  — l. 
that  is,  these  conditions  being  satisfied,  the  equation  — 1 = 0 requii’ed  to  be  satisfied, 
will  be  identical  with  the  equation  which  is  always  satisfied,  and  wall  therefore  itself 
be  satisfied.  And  in  like  manner  the  matrix  M of  the  order  2 mil  satisfy  the  condition 
M®— 1 = 0,  or  will  be  periodic  of  the  third  order,  if  only  M®— 1 contams  as  a factor 
M.^—[a-{-d)yL-\-ad—hc, 
and  so  on. 
31.  But  suppose  it  is  required  to  find  a matrix  of  the  order  3, 
M = ( a,  c ) 
d,  e,  f 
g,  h,  i 
which  shall  be  periodic  of  the  second  order. 
a — M,  h , c : 
d , e— M,  / 
g , h , M 
the  matrix  here  satisfies 
Writing  for  shortness 
-(M^-AMHBM-C), 
M^-AjVF+BM-C=0, 
and,  as  before,  the  assumed  equation  is  1 = 0.  Here,  if  we  have  1-|-B=0,  x^-j-C=0, 
the  left-hand  side  will  contain  the  factor  (M^ — 1),  and  the  equation  will  take  the  form 
(M^  — l)(M-j-C)=0,  and  we  should  have  then  M- — 1 = 0,  provided  M-|-C  were  not  an 
indeterminate  matrix.  But  M-ffiC  denotes  the  matrix 
( a-fC,  b , c ) 
d , ^+C,  f j 
g , A , '(t+C  I 
the  determinant  of  which  is  C^fi-AC^-j-BC  + C,  which  is  equal  to  zero  in  virtue  of 
