Remark

(1) In Chapter III, we shall see that, if $m = n = r$ and $K$ is commutative, then, for all choices of $P$ and $Q$ satisfying the conditions of Proposition 14, the element $\delta_r$ is always the same and equal to the determinant of $X$ (III, § 8, no. 6).

(2) The argument of Proposition 14, slightly modified, shows that there is a permutation matrix $R$ such that (with the same conditions on $P$)

if $m = n = r$ does not hold, and

otherwise. Observe also that the method of proof gives an explicit determination of the matrices $P, Q, R$ when $X$ is given explicitly.