676 Proceedings of the Royal Society 
Consider the matrix M = ( a, b ) ; write down the equation, 
1 «, I 
1 a - M, b 1=0, 
I c ,d - M I 
where the function on the left hand is a determinant, M being 
therein regarded in the first instance as a quantity, viz., this equa- 
tion is 
M 2 - (a + d) M + ( ad - be) M° = 0 ; 
and then substituting for M 2 , M, M°, their expressions as matrices, 
this equation is identically true, viz., it stands for the four iden- 
tities — 
a 2 + be - (a + d) a + ad - be = 0, 
b(a + d) - (a + d) b ,= 0, 
c(a + d) — (a + d) c =0, 
d 2 + be - (a + d) d + ad - be - 0, 
and the like property holds for a matrix of any order. 
To extract the square root of the matrix M = ( a, b ) ; in 
I c , d \ 
other words, to find a matrix L = ( a, b ) such that L 2 = M; 
| 0, d [ 
that is 
( a 2 -(- be, b(a + d) ) = ( a, b ) , 
| c(a + d), d 2 + be | | c, d j 
(four equations for the determination of a, b, c, d) : — 
The solution is as follows : write 
I a - M, b I = M 2 - pM + q , 
I c , d - M | 
is here written for g-M 0 , and so in other cases) ; and similarly 
| a - L, b I = L 2 - pL -f q, 
j c , d - L I 
then we have 
M 2 - + <£=0, 
L 2 - pL + q = 0, 
L2 = M; 
and from these equations we may express L as a linear function of 
M, M°, with coefficients depending on p, q; and also determine 
the unknown quantities p, q in terms of p, q. 
