22 
ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
in  fact  this  equation  gives 
{x,  y,z)=(  A,  A',  A" 
IX,  Y,  Z)=(( 
a , b , c 
‘ 
B,  B',  B" 
a!  , b'  , d 
C,  C',  C" 
a!’,  b\  d' 
and  we  have,  for  the  determination  of  the  coefficients  of  the  inverse  or  reciprocal  matrix, 
the  equations 
{ A,  M,  A!’  X a , b , c )=(  1,  0,  0 ), 
B,  B',  B" 
C,  C',  C" 
a'  , h'  , d 
a”,  b",  d' 
0,  1,  0 
0,  0,  1 
( a , b , c X A,  A',  A"  ) = ( 1,  0,  0 ), 
a'  , b'  , d 
a",  b\  d' 
B,  B',  B"  I 0,  1,  0 
C,  C',  C"  0,  0,  1 
which  are  equivalent  to  each  other,  and  either  of  them  is  by  itself  sufficient  for  the  com- 
plete determination  of  the  inverse  or  reciprocal  matrix.  It  is  well  known  that  if  V 
denote  the  determinant,  that  is,  if 
\/  — (L  ^ b ^ c 
a',  V , d 
b\  d' 
then  the  terms  of  the  inverse  or  reciprocal  matrix  are  given  by  the  equations 
A=i 
V 
1,  0,  0 
0,  b',  d 
0,  b”,  d' 
B= 
0 , 1,  0 , &c. 
a' , 0,  d 
a”,  0,  d' 
or  what  is  the  same  thing,  the  inverse  or  reciprocal  matrix  is  given  by  the  equation 
( a , b , c )-'  _1_(  b„,V,  ) 
^4^5  I 
b„V,  1 
a',  b',  d 1 
V 
b\  d' ; 
where  of  course  the  differentiations  must  in  every  case  be  performed  as  if  the  terms  a,  b, 
See.  were  all  of  them  independent  arbitrary  quantities. 
18.  The  formula  shows,  what  is  indeed  clear  d priori,  that  the  notion  of  the  inverse 
or  reciprocal  matrix  fails  altogether  when  the  determinant  vanishes;  the  matrix  is  in 
this  case  said  to  be  indeterminate,  and  it  must  be  understood  that  in  the  absence  of 
express  mention,  the  particular  case  in  question  is  frequently  excluded  from  considera- 
tion. It  may  be  added  that  the  matrix  zero  is  indeterminate ; and  that  the  product  of 
two  matrices  may  be  zero,  without  either  of  the  factors  being  zero,  if  only  the  matrices 
are  one  or  both  of  them  indeterminate. 
