ME.  A.  CATLET  ON  THE  THEOEY  OE  MATEICES. 
27 
so  that  M satisfies  the  equation 
a — M,  h j = 0, 
c , M I 
or 
and  let  X^,  be  the  single  quantities,  roots  of  the  equation 
«-X,  b 
c , d-^ 
= 0 
or 
X*— («4-f?)X+a(7— Je=0. 
The  equation  satisfied  by  the  matrix  may  be  written 
(M-XJ(M-XJ=0, 
in  which  X^  X,^  are  to  be  considered  as  respectively  invohing  the  matrix  unity,  and  it 
would  at  first  sight  seem  that  we  ought  to  have  one  of  the  simple  factors  equal  to  zero, 
which  is  obviously  not  the  case,  for  such  equation  would  signify  that  the  perfectly  inde- 
terminate matrix  M was  equal  to  a single  quantity,  considered  as  involving  the  matrix 
unity.  The  explanation  is  that  each  of  the  simple  factors  is  an  indeterminate  matrix, 
in  fact  M— X^  stands  for  the  matrix 
( a-X„  5 ), 
I c , d—X.1 
and  the  determinant  of  this  matrix  is  equal  to  zero.  The  product  of  the  two  factors  is 
thus  equal  to  zero  without  either  of  the  factors  being  equal  to  zero. 
29.  A matrix  satisfies,  we  have  seen,  an  equation  of  its  own  order,  involving  the 
coefficients  of  the  matrix;  assume  that  the  matrix  is  to  be  determined  to  satisfy  some 
other  equation,  the  coefficients  of  which  are  given  single  quantities.  It  would  at  first 
sight  appear  that  we  might  eliminate  the  matrix  between  the  two  equations,  and  thus 
obtain  an  equation  which  would  be  the  only  condition  to  be  satisfied  by  the  terms 
of  the  matrix ; this  is  obviously  wrong,  for  more  conditions  must  be  requisite,  and  w^e 
see  that  if  we  were  then  to  proceed  to  complete  the  solution  by  finding  the  value  of  the 
matrix  common  to  the  twm  equations,  w’e  should  find  the  matrix  equal  in  every  case 
to  a single  quantity  considered  as  involving  the  matrix  unity,  which  it  is  clear  ought 
not  to  be  the  case.  The  explanation  is  similar  to  that  of  the  difficulty  before  adverted 
to,  the  equations  may  contain  one,  and  only  one,  common  factor,  and  may  be  both  of 
them  satisfied,  and  yet  the  common  factor  may  not  vanish.  The  necessary  condition 
seems  to  be,  that  the  one  equation  should  be  a factor  of  the  other ; in  the  case  where 
the  assumed  equation  is  of  an  order  equal  or  superior  to  the  matrix,  then  if  this  equation 
contain  as  a factor  the  equation  which  is  always  satisfied  by  the  matrix,  the  assumed 
equation  wfill  be  satisfied  identically,  and  the  condition  is  sufficient  as  well  as  necessary : 
E 2 
