ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
25 
number  of  unknown  quantities  from  (if  n be  the  order  of  the  matrix)  down  to  n.  To 
complete  the  solution,  it  is  necessary  to  compare  the  value  obtained  as  above,  with  the 
assumed  value  of  the  irrational  function,  which  will  lead  to  equations  for  the  determina- 
tion of  the  n unknown  quantities. 
26.  As  an  illustration,  consider  the  given  matrix 
M=(  a,  d ) 
c,  d I 
and  let  it  be  requhred  to  find  the  matrix  L=\/M.  In  this  case  M satisfies  the  equation 
and  in  like  manner  if 
L=(  a,  ) 
then  L satisfies  the  equation 
— (a-|-^)L-}-a§ — |3y=0  ; 
and  from  these  two  equations,  and  the  rationalized  equation  L^=M,  it  should  be  possible 
to  express  L in  the  form  of  a linear  function  of  M : in  fact,  putting  in  the  last  equation 
for  its  value  (=M),  we  find  at  once 
L= 
which  is  the  required  expression,  invoking  as  it  should  do  the  coefficients  «+§,  — |3y 
of  the  equation  in  L.  There  is  no  difficulty  in  completing  the  solution ; write  for  short- 
ness a+§=X.  aS— /3y=Y,  then  we  have 
L=(  (3  ) = ( « + Y 
^ ' X 
C 
X 
X 
rf+Y 
and  consequently  forming  the  values  of  a+S  and  ab — /3y, 
Q,  d-\-  2Y 
X: 
Y = 
X 
[fl-l- Y)  {d+Y)—bc 
X2 
and  putting  also  a-\-d=Y,  ad  — hc=zQ,^  we  find  without  difficulty 
X=x/P+2yQ, 
Y=v/Q, 
and  the  values  of  a,  j3,  y,  b are  consequently  known.  The  sign  of  \/ Q is  the  same  in 
both  formulae,  and  there  are  consequently  in  all  four  solutions,  that  is,  the  radical 
has  four  values. 
MDCCCLVIII. 
E 
