755] 
ON A MATRIX AND A FUNCTION CONNECTED THEREWITH. 
253 
means the two equations x x — ax + by, y 1 = cx + dy; and then if x 2 , y 2 are derived in 
like manner from x 1} y 1} that is, if x 2 = ax 1 -\-by 1 , y 2 = cx 1 + dy 1} and so on, x n , y n will 
be linear functions of x, y; say we have x n — a n x + b n y, y n = c n x + d n y: and the nth. 
power of ( a, b ) is, in fact, the matrix (a», b n ). 
I ^ I | Cjj, , d n | 
In particular, we have 
( a, b ) 2 , =( a 2 , b 2 ), =( a 2 + bc, b(a + d) ), 
c> j | c 2 , d 2 | j c (a -f- d), d 2 4- be \ 
and hence the identity 
( a, b ) 2 — (a+d)( a, b ) + (ad-bc)( 1, 0 ) = 0; 
| c, d | c, d 0, 1 
viz. this means that the matrix 
( ci 2 -(a + d) a + ad-be, b 2 - (a + d)b ) = ( 0, 0 ), 
| c 2 — (a + d) c , d 2 —(a + cl) d + ad — be \ j 0, 0 I 
or, what is the same thing, that each term of the left-hand matrix is =0; which is 
at once verified by substituting for a 2 , b 2 , c 2 , d 2 their foregoing values. 
The explanation just given will make the notation intelligible and show in a 
general way how a matrix may be worked in like manner with a single quantity: 
the theory is more fully developed in my Memoir above referred to. I proceed 
with the solution in the algorithm of matrices. Writing for shortness M=( a, b ), 
| j 
I c, d j 
the identity is 
M 2 — (a + d) M + (ad — be) = 0, 
the matrix (1,0) being in the theory regarded as = 1; viz. M is determined by 
| o, 1 ! 
a quadric equation; and we have consequently M n = a linear function of M. Writing 
this in the form 
M n -AM + B = 0, 
the unknown coefficients A, 
of the equation 
viz. we have 
B can be at once obtained in terms of a, /3, the roots 
u 2 — (a + d)u + ad — be = 0, 
a n -Aa+B = 0, 
/3 n - A/3 + B = 0; 
or more simply from these equations, and the equation for M n , eliminating a, /3, we 
have 
M n , 
M, 
d n , 
a , 
/3\ 
P,