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Transactions of the Boyal Society of South Africa. 
co-factor in tlie determinant being — 4d, and the value of the determinant 
itself being consequently — 16d/'^ 
When is a complex quantity whose modulus is 1, the determinant is 
both inversely orthogonal and equimodular, and therefore is a determinant 
of maximum value. This was first pointed out by Hadamard. 
14. If in addition to requiring the elements of fx to be unimodular we 
insist on them being real- — in other words, if we seek to construct maxi- 
mum determinants whose elements are +1 or — 1 — we soon find that the 
problem is soluble only for certain orders of determinants. We can show, 
however, that if a solution be obtained for order r it is easy to give a solu- 
tion for order 2r. For the rows of the r-line determinant being A, B, C, ... 
we know that 
AB = 0, AC = 0, AD=-0, 
BC = 0, BD = 0, 
CD = 0, 
and this being the case the 2r-line determinant whose rows are 
(A, A), (A, -A), (B, B), (B, -B), (C, C), (C, -C), 
has evidently the same property. Thus, the determinant for the 2nd 
order being 
the determinant for the 4th order is 
1| 
1 , 
1 
- 1 
- 1 
1 
which agrees with the result of § 13. 
The determinants of order 2™ thus obtainable are all axisymmetric. 
15. Hadamard' s 12-line determinant of this kind has also a latent 
axisymmetry which it is preferable to put in evidence. If we denote a 
row of three elements by the place-numbers of those which are negative, 
thus 
1 1 -1 by 3, -1 1 -1 by 13, 
* Besides the solution obtained in this paragraph there are at least two others, the set 
of three being 
a, b, c, d, e, f=l, —1, -1, x, —x,x, 
— 00 y X J «X ) 1 J 1 y 00 • 
They are not, however, really different. 
