MR.  A.  CAYLEY  OX  THE  THEORY  OE  MATRICES. 
which  shows  that  a matrix  compounded  with  the  transposed  matrix  gives  rise  to  a S}Tn- 
metrical  matrix.  It  does  not  however  follow,  nor  is  it  the  fact,  that  the  matrix  and 
transposed  matrix  are  convertible.  And  also 
d 
c,  d 
b,  d 
+ dcd  -\-h{a^-\-d-). 
c^-\-abd-\-c{ar-\-d^)  ) 
d^-\-abc-{-d(b^-\-(f)  I 
which  is  a remarkably  symmetrical  form.  It  is  needless  to  proceed  further,  since  it  is 
clear  that 
( «,  c X ^ X I ^ ) = ((«,<?  X ^ 
b,  d 
d 
b,  d 
c,  d 
b,  d 
c\  d 
44.  In  all  that  precedes,  the  matrix  of  the  order  2 has  frequently  been  considered, 
but  chiefly  by  way  of  illustration  of  the  general  theory ; but  it  is  worth  while  to  develope 
more  particularly  the  theory  of  such  matrix.  I call  to  mind  the  fundamental  properties 
which  have  been  obtained,  viz.  it  was  shown  that  the  matrix 
satisfies  the  equation 
and  that  the  two  matrices 
will  be  convertible  if 
M=('  b ), 
c,  d I 
'W' d)yi-\-ad — ^c=0, 
( a,  b { a',  b'  ), 
c,  d \ I c',  d' 
al  — d' : b' : c=a — d :b  \ c, 
and  that  they  will  be  skew  convertible  if 
«+<^=0,  a!-\-d'=^,  ac^  -{-bd  -\-b'c-\-dd^  — 
the  first  two  of  these  equations  being  the  conditions  in  order  that  the  two  matrices  may 
be  respectively  periodic  of  the  second  order  to  a factor  pres. 
45.  It  may  be  noticed  in  passing,  that  if  L,  M are  skew  convertible  matrices  of  the 
order  2,  and  if  these  matrices  are  also  such  that  L^=  — I,  M^=  — 1,  then  putting 
N=LM=:  — ML,  we  obtain 
L-*=-I,  M^=-I,  N=*=-I, 
L=MN=-NM,  M=NL=-NL,  N=LM=-ML, 
which  is  a system  of  relations  precisely  similar  to  that  in  the  theory  of  quaternions. 
46.  The  integer  powers  of  the  matrix 
M=(  «,  b ), 
c,  d 
are  obtained  with  great  facility  from  the  quadratic  equation ; thus  we  have,  attending 
first  to  the  positive  powers, 
