478 
A MEMOIR ON THE THEORY OF MATRICES. 
[152 
7. It is clear that we have L + M = M + L, that is, the operation of addition is 
commutative, and moreover that (L + M) + N= L + (M + N) = L + M + X, that is, the 
operation of addition is also associative. 
8. The equation 
(X, Y, Z) = ( a , b , c Jinx, my, mz) 
a', V, c' 
a", b", c" 
written under the forms 
(X, Y, Z) = m( a , b , c ]£#, y, z) = ( via , mb , me \x, y, z) 
a', b', c' ma', mb', me' 
gives 
a”, 
b", 
c" 
ma", 
mb", 
m ( 
a , 
b , c 
)=( 
ma , 
mb , 
me 
a', 
b' 
c' 
ma', 
mb', 
me' 
a", 
b" 
c" 
ma", 
mb", 
me" 
as the rule for the multiplication of a matrix by a single quantity. The multiplier m 
may be written either before or after the matrix, and the operation is therefore com 
mutative. We have it is clear m{L + M) = mL + viM, or the operation is distributive. 
9. The matrices L and viL may be said to be similar to each other ; in 
particular, if m = 1, they are equal, and if m = — 1, they are opposite. 
10. We have, in particular, 
m ( 1, 0, 0 ) = ( m, 0, 0 ), 
o 
i—i 
o 
o 
o 
0, 0, 1 
© 
© 
§ 
or replacing the matrix on the left-hand side by unity, we may write 
vi — (m, 0, 0 ); 
0, m, 0 
0, 0, m 
the matrix on the right-hand side is said to be the single quantity m considered as 
involving the matrix unity. 
11. The equations 
(X, Y, Z) = { a , b , c Joe, y, z), {x, y, z) = ( a , /3 , y $£ y, £), 
a', b', c' 
/3', y 
a", b", c" 
/3", l"