A Theorem of Frobenius’. 
345 
1921-22.] 
in it a single row of Q , 2 A 2 the similar sum got by zero-ising in A two 
rows of Q, and so on, then 
is equal to the determinant got from A by zero-ising P. For example, 
taking n = 5, r = 2, s = S, the determinant as partitioned being thus 
h 
«2 
h 
CO CO 
«4 
b. 
“5 
h 
^3 
^■4 
^5 
d. 
d. 
^1 
^2 
^4 
^5 
we have l 
«2 
- 
«2 
«3 
a, 
«5 
+ 
«i 
«2 
^3 
bi 
b. 
h 
b, 
^5 
bs 
^2 
h 
h 
b. 
b^ 
^2 
^3 
«4 
'•s 
^2 
^'3 
0 
^‘5 
^1 
^'2 
*^3 
^4 
d, 
do 
ds 
d^ 
d. 
d. 
d. 
dz 
d. 
A 
d. 
d. 
dz 
d. 
^2 
^3 
^4 
^1 
62 
^3 
«4 
65 
«2 
^3 
64 
^5 
= 
A 
^'1 
^2 
^'3 
% 
do 
^3 
d. 
ds 
63 
^3 
64 
^5 
This is readily verified by expanding its five determinants in terms of the 
2-line minors of their last two columns and the complementary minors. 
The number of determinants on the left of the general equality is 
evidently 
1 -f r 4 - Jr(r - 1) + .... 
(6) Our next point is that the principle involved in the theorem of the 
preceding paragraph extends beyond the sphere of Frobenius’ theorem, the 
places liable to be filled with zeros being not necessarily those of a 
rectangular array. They may be, for example, the places of the principal 
diagonal, for we have the theorem that — If A be any n-line determinant 
and Ar any determinant got from it by deleting r diagonal elements, then 
A-2,^i+2j^2“ • • • • 
is equal to the diagonal term. Thus 
«1 
«2 «3 
^^2 ^3 
- z 
^2 ^3 
l\ ^^2 ^3 
+ z 
. «2 «3 
• ^3 
- 
. (X3 
il . *2 
^2 ^3 
Cl C2 C3 
Cl C2 C3 
Cl C2 
