ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
33 
M?=.[_{a-\-df—{ad—bcy\'M.—{a-\-d){ad—bc), 
&c.,  • 
whence  also  the  conditions  in  order  that  the  matrix  may  be  to  a factor  periodic  of 
the  orders  2,  3,  &c.  are 
a-\-d  =0, 
{a-\-df—{ad—bc)—^, 
&c. ; 
and  for  the  negative  powers  we  have 
{ad—bc)M.~'=i  — 
which  is  equivalent  to  the  ordinary  form 
Jc)M~‘  = ( d,  —b  ); 
I — c,  a 
and  the  other  negative  powers  of  M can  then  be  obtained  by  successive  multiplications 
with  M~', 
47.  The  expression  for  the  %th  power  is  however  most  readily  obtained  by  means  of 
a particular  algorithm  for  matrices  of  the  order  2. 
Let  h,  b,  c,  J,  q be  any  quantities,  and  wite  for  shortness  E.=  — Id — ^bc  \ suppose  also 
that  A',  i',  (f,  J',  are  any  other  quantities,  such  nevertheless  that  A' : V : c'=  h:b:  c,  and 
write  in  like  manner  R'=  — A'^— 45'c'.  Then  observing  that  respect- 
N V d 
ively  equal  to  -7=  — the  matrix 
^ vR'’  vR'  vR' 
contains  only  the  quantities  J,  q,  which  are  not  the  same  in  both  systems ; and  we  may 
therefore  represent  this  matrix  by  (J,  q),  and  the  corresponding  matrix  with  h\  b',  d,  J',  q' 
by  (J',  The  two  matrices  are  at  once  seen  to  be  convertible  (the  assumed  relations 
h!  \ b' : d —h\b  \ c correspond  in  fact  to  the  conditions,  a! — d'  :b’ : d~a — d:  b : c,  of  con- 
vertibility for  the  ordinary  form),  and  the  compound  matrix  is  found  to  be 
/sin  jq  + q')  A 
And  in  like  manner  the  several  convertible  matrices  (J,  q),  (J',  q'),  (J",  q")  &c.  give 
the  compound  matrix 
sin  {q  + q'  + q"..) 
sin  qsin  q'  sin  q" .. 
MDCCCLVIII. 
F 
