ME.  A.  CAYLEY  ON  THE  THEOEY  OF  MATEICES. 
35 
50.  The  matrix  will  be  to  a factor  periodic  of  the  ni\i  order  if  only  sin  wi^=0, 
that  is,  if  2^=  ^ (w  must  be  prime  to  w,  for  if  it  were  not,  the  order  of  periodicity  would  be 
not  n itself,  but  a sub  multiple  of  n) ; but  cos  §'==  ^ condition  is  therefore 
rmr 
:0, 
{d-\-af—^{ad— he)  co^  ^ 
or  as  this  may  also  be  written, 
d‘^-\-a^—2ad  cos^^-f-4^c  cos^— = 0, 
a result  which  agrees  with  those  before  obtained  for  the  particular  values  2 and  3 of 
the  index  of  periodicity. 
51.  I may  remark  that  the  last  preceding  investigations  are  mtijnately  connected  with 
the  investigations  of  Babbage  and  others  in  relation  to  the  function  (px-=  — —i' 
I conclude  with  some  remarks  upon  rectangular  matrices. 
52.  A matrix  such  as 
i a,  h,  c ) 
I V,  c'  \ 
where  the  number  of  columns  exceeds  the  number  of  lines,  is  said  to  be  a broad  matrix ; 
a matrix  such  as 
( a , b ) 
a',  b' 
a”,  b" 
where  the  number  of  lines  exceeds  the  number  of  columns,  is  said  to  be  a deep  matrix. 
53.  The  matrix  zero  subsists  in  the  present  theory,  but  not  the  matrix  unity. 
Matrices  may  be  added  or  subtracted  when  the  number  of  the  lines  and  the  number  of 
the  columns  of  the  one  matrix  are  respectively  equal  to  the  number  of  the  hues  and  the 
number  of  the  columns  of  the  other  matrix,  and  under  the  hke  condition  any  number 
of  matrices  may  be  added  together.  Two  matrices  may  be  equal  or  opposite  the  one  to 
the  other.  A matrix  may  be  multiplied  by  a single  quantity,  giving  rise  to  a matrix  ot 
the  same  form ; two  matrices  so  related  are  similar  to  each  other. 
54.  The  notion  of  composition  applies  to  rectangular  matrices,  but  it  is  necessary  that 
the  number  of  lines  in  the  second  or  nearer  component  matrix  should  be  equal  to  the 
number  of  columns  in  the  first  or  further  component  matrix ; the  compound  matrix  will 
then  have  as  many  lines  as  the  first  or  further  component  matrix,  and  as  many  columns 
as  the  second  or  nearer  component  matrix. 
55.  As  examples  of  the  composition  of  rectangular  matrices,  we  have 
(a,b,cX  C,  d’  ) = ((«,  b,  cX«',  e',  i%  («,  b,  cXh\fJ’){(i.  h cXc\  g',  Jd),  («,  b,  c^d’,  //,  Z') ), 
(Z,  e,f\ 
A' 
Z' 
(cZ,  d,  ^'),  {d,  e,fXb'J\f)(d,  eJXc\  g\  {d,  e,fXd-'^  h\  Z') 
F 2 
