152] 
A MEMOIR ON THE THEORY OF MATRICES. 
4 77 
3. The quantities (X, Y, Z) will be identically zero, if all the terms of the matrix 
are zero, and we may say that 
(0, 0, 0 ) 
0, 0, 0 
0, 0, 0 
is the matrix zero. 
Again, (X, F, Z) will be identically equal to (x, y, z), if the matrix is 
(1, 0, 0 ) 
0, 1, 0 
0, 0, 1 
and this is said to be the matrix unity. We may of course, when for distinctness it 
is required, say, the matrix zero, or (as the case may be) the matrix unity of such an 
order. The matrix zero may for the most part be represented simply by 0, and the 
matrix unity by 1. 
4. The equations 
a', b' , c' 
a", b", c" 
a", /3", 7" 
give 
(A + X.', Y+ Y, Z + Z) — ( a + a , b + /3 , c + y Qoc. y, z) 
a + a. , b' + /3', c' + y 
a" + a", &" + £", c" + y" 
and this leads to 
( a + a , b + /3 , c + 7 
) = ( 
a , b , c )+( 
a » £ , 7 ) 
a + a' , b' + /3', c' + 7' 
a', b', c' 
/3', 7' 
a" + a", b"+t3", c" + 7" 
a", b", c" 
/3", 7" 
as a rule for the addition of matrices; that for their subtraction is of course similar 
to it. 
5. A matrix is not altered by the addition or subtraction of the matrix zero, 
that is, we have M ± 0 = M. 
The equation L = M, which expresses that the matrices L, M are equal, may also 
be written in the form L — M = 0, i. e. the difference of two equal matrices is the 
matrix zero. 
6. The equation L = — M, written in the form L + M = 0, expresses that the sum 
of the matrices L, M is equal to the matrix zero, the matrices so related are said to be 
opposite to each other; in other words, a matrix the terms of which are equal but oppo 
site in sign to the terms of a given matrix, is said to be opposite to the given matrix.