ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
21 
as  the  rule  for  the  multiplication  or  composition  of  two  matrices.  It  is  to  be 
observed,  that  the  operation  is  not  a commutative  one;  the  component  matrices  may 
be  distinguished  as  the  first  or  further  component  matrix,  and  the  second  or  nearer 
component  matrix,  and  the  rule  of  composition  is  as  follows,  viz.  any  line  of  the  com- 
pound matrix  is  obtained  by  combining  the  corresponding  line  of  the  first  or  further 
component  matrix  successively  with  the  several  columns  of  the  second  or  nearer  com- 
pound matrix. 
12.  A matrix  compounded,  either  as  first  or  second  component  matrix,  with  the  matrix 
zero,  gives  the  matrix  zero.  The  case  where  any  of  the  terms  of  the  given  matrix  are 
infinite  is  of  course  excluded. 
13.  A matrix  is  not  altered  by  its  composition,  either  as  first  or  second  component 
matrix,  with  the  matrix  unity.  It  is  compounded  either  as  first  or  second  component 
matrix,  with  the  single  quantity  m considered  as  involving  the  matrix  unity,  by  multi- 
plication of  all  its  terms  by  the  quantity  m : this  is  in  fact  the  before-mentioned  rule 
for  the  multiplication  of  a matrix  by  a single  quantity,  which  rule  is  thus  seen  to  be  a 
particular  case  of  that  for  the  multiplication  of  two  matrices. 
14.  We  may  in  like  manner  multiply  or  compound  together  three  or  more  matrices : 
the  order  of  arrangement  of  the  factors  is  of  course  material,  and  we  may  distinguish 
them  as  the  first  or  furthest,  second,  third,  &c.,  and  last  or  nearest  component  matrices, 
but  any  two  consecutive  factors  may  be  compounded  together  and  replaced  by  a single 
matrix,  and  so  on  until  all  the  matrices  are  compoimded  together,  the  result  being  inde- 
pendent of  the  particular  mode  in  which  the  composition  is  efiected ; that  is,  we  have 
L.MN=LM.N=LMN,  LM.NP=L.MN.P,  &c.,  or  the  operation  of  multiplication, 
although,  as  already  remarked,  not  commutative,  is  associative. 
15.  We  thus  arrive  at  the  notion  of  a positive  and  integer  power  If  of  a matrix  L, 
and  it  is  to  be  observed  that  the  different  powers  of  the  same  matrix  are  convertible. 
It  is  clear  also  that^  and  being  positive  integers,  we  have  lf.lf=ll'^'\  which  is  the 
theorem  of  indices  for  positive  integer  powers  of  a matrix. 
16.  The  last-mentioned  equation,  L^.L’=L^’*'^,  assumed  to  be  true  for  all  values  what- 
ever of  the  indices  p and  q,  leads  to  the  notion  of  the  powers  of  a matrix  for  any  form 
whatever  of  the  index.  In  particular,  L^.L®=:L^  or  L®=1,  that  is,  the  0th  power  of  a 
matrix  is  the  matrix  unity.  And  then  putting ^ = 1,  q=  — \,  or  p=  — l,  q=l,  we  have 
L.L“'  = L“bL=l ; that  is,  L"',  or  as  it  may  be  termed  the  inverse  or  reciprocal  matrix, 
is  a matrix  which,  compounded  either  as  first  or  second  component  matrix  with  the 
original  matrix,  gives  the  matrix  unity. 
17.  We  may  arrive  at  the  notion  of  the  inverse  or  reciprocal  matrix,  directly  from  the 
equation 
(X,  Y,  Z)=('  a , h , c y,  z), 
a!,  h\  c' 
a!’,  h\  f 
