26 
ME.  A.  CAYLEY  ON  THE  THEOEY  OF  JMATEICES. 
27.  To  illustrate  this  further,  suppose  that  instead  of  M we  have  the  matrix 
M^=(  a,  h )^=(  (f-\-'bc  , 5(a+(7)  ), 
1 c,  1 I c{a-\-d\  d^-^-hc  1 
so  that  L^=]Vr,  we  find 
Y—{a-\-d'f—1{€Ld—lc\ 
Q—[ad—hcf^ 
and  thence  ^/Q=±,{ad—bc).  Taking  the  positive  sign,  we  have 
and  these  values  give  simply 
But  taking  the  negative  sign. 
Y =.ad — he, 
X=  + («+(^), 
L=  + ( a,  b ) = ±M, 
1 c,  I 
Y =-—ad-\-bc, 
X=+V^  {a—df-\-iibc, 
and  retaining  X to  denote  this  radical,  we  find 
L=(^  a^—ad+2bc  b{a->i-d)  ), 
X ’ X 
c{a-\-d)  d'^  — ad+2bc 
X ’ “~X 
which  may  also  be  written 
y_a  + «?  ( a,  b ^ 2 {ad— be)  ( 1,  0 }, 
~ ^ I ^ 1 “ X I n 1 I 
\ c,  d I 0,  1 I 
or,  what  is  the  same  thing, 
-r  a + d^r  2{ad—bc) 
and  it  is  easy  to  verify  d posteriori  that  this  value  in  fact  gives  L*=!VP.  It  may  be 
remarked  that  if 
M^=(  1,  0 )^=1, 
I 0,  1 I 
the  last-mentioned  formula  fails,  for  we  have  X=0;  it  will  be  seen  presently  that  the 
equation  L^=l  admits  of  other  solutions  besides  L=+l.  The  example  shows  how  the 
values  of  the  fractional  powers  of  a matrix  are  to  be  investigated. 
28.  There  is  an  apparent  difficulty  connected  with  the  equation  satisfied  by  a matrix, 
which  it  is  proper  to  explain.  Suppose,  as  before, 
M=(  a,  b ), 
