in which the elements fulfil the relation fh k = -fh k, is called an alternating matrix. These relations say that the transposed matrix ḟ = -f. Then by f* will be the dual, alternating matrix
(35)
f* = | 0 f₃₄ f₄₂ f₂₃ |
| f₄₃ 0 f₁₄ f₃₁ |
| f₂₄ f₄₁ 0 f₁₂ |
| f₃₂ f₁₃ f₂₁ 0 |
Then (36) f* f = f₃₄ f₂₂ + f₄₂ f₃₁ + f₃₂ f₂₄
i.e. We shall have a 4 × 4 series matrix in which all the elements except those on the diagonal from left up to right down are zero, and the elements in this diagonal agree with each other, and are each equal to the above mentioned combination in (36).
The determinant of f is therefore the square of the combination, by Det½f we shall denote the expression
Det½f