[ 17  ] 
II.  A Mmioir  on  the  Thecyry  of  Matrices.  By  Aethue  Cayley,  Esq.,  F.R.S. 
Eeceived  December  10,  1857, — Bead  January  14,  1858. 
The  term  matrix  might  be  used  in  a more  general  sense,  but  in  the  present  memoir  I 
consider  only  square  and  rectangular  matrices,  and  the  term  matrix  used  without  quali- 
fication is  to  be  understood  as  meaning  a square  matrix ; in  this  restricted  sense,  a set 
of  quantities  arranged  in  the  form  of  a square,  e.  g. 
(a,h,c) 
a! , V , d 
d',  b",  d' 
is  said  to  be  a matrix.  The  notion  of  such  a matrix  arises  naturally  from  an  abbreviated 
notation  for  a set  of  linear  equations,  viz.  the  equations 
'K.z=ax  -\-by  +cz, 
Y =a'x  -\-b'y  -\-dz, 
Z =a''a:-\-b"y-{-(^'z, 
may  be  more  simply  represented  by 
(X,  Y,  Z)=(  a , b , c Jx,y,  z), 
a',  b' , d 
a",  b\  c" 
and  the  consideration  of  such  a system  of  equations  leads  to  most  of  the  fundamental 
notions  in  the  theory  of  matrices.  It  will  be  seen  that  matrices  (attending  only  to  those 
of  the  same  order)  comport  themselves  as  single  quantities;  they  may  be  added, 
multiplied  or  compounded  together,  &c. : the  law  of  the  addition  of  matrices  is  pre- 
cisely similar  to  that  for  the  addition  of  ordinary  algebraical  quantities ; as  regards  their 
multiplication  (or  composition),  there  is  the  peculiarity  that  matrices  are  not  in  general 
convertible ; it  is  nevertheless  possible  to  form  the  powers  (positive  or  negative, 
integral  or  fractional)  of  a matrix,  and  thence  to  arrive  at  the  notion  of  a rational  and 
integral  function,  or  generally  of  any  algebraical  function,  of  a matrix.  I obtain  the 
remarkable  theorem  that  any  matrix  whatever  satisfies  an  algebraical  equation  of  its 
own  order,  the  coefficient  of  the  highest  power  being  unity,  and  -those  of  the  other 
powers  functions  of  the  terms  of  the  matrix,  the  last  coefficient  being  in  fact  the  deter- 
minant ; the  rule  for  the  formation  of  this  equation  may  be  stated  in  the  following  con- 
densed form,  which  will  be  intelligible  after  a perusal  of  the  memoir,  viz.  the  determi- 
MDCCCLVIII.  D 
