@Manny-Reyes: i know that $\mathbf{AA}^\mathrm D = \mathbf A^\mathrm D \mathbf A = (\det \mathbf A) \mathbf I$ which gives our result if $\mathbf A$ is invertible, because then $\mathbf{AA}^{-1} = \mathbf A^{-1} \mathbf A = \mathbf I$.

i'll check it out right away—if it's possible would you be kind enough to explain in plain language (basic linear algebra if it's possible) how does action of exterior power works? :) thanks in advance for your effort!

hmm… i know for example that matrix transpose corresponds to dualization functor or rather it is it's special case. for transpose i know that it's functorial: $I^\mathrm T = I$, $(\mathbf{AB})^\mathrm T = \mathbf B^\mathrm T \mathbf A^\mathrm T$ and it commutes with inverse. this functor is $(\cdot)^\star\colon \mathrm{Hom}(M, N) \to \mathrm{Hom}\left(N^\star, M^\star\right)$ for $R$-modules $M$ and $N$, where $M^\star, N^\star$ are their dual modules respectively, and $\mathrm{Hom}(M, N)$ denotes set of all homomorphisms $M \to N$. i'm looking for similar explanation! :p

i think you've helped me quite much, so please excuse my overfamiliar and presumptuous behaviour—i hadn't in mind giving orders to anybody: i just don't feel the courtesy and politeness levels of the language good enough yet.<br>mind to explore the topic little further as you proposed? :)

can i say then that minor of 'codegree' 1 describes measure of parallelotope's face of codimension 1 spaned by appropriate vectors? if so, what can i say about other minors in similar manner—what are principal minors then? (i'm afraid that hodge duality might spring out… ;p) tell me how to interpret laplace formula where minors show up (i won't have problems with cofactors—signs seem to come from keeping track of orientation…) probably your answer is definite but i'm still looking here for purely geometrical description which i could understand—care to give completely new answer? :p

i just forgot about exterior algebra… :p i just don't get it thoroughly but as far as i get it: the question i posed in your language would sound like 'how does $T$ it act on those parallelotopes…?' could you dwell little bit more on this? i must confess i'd be more satisfied seeing more school geometry here—care to translate from exterior algebra? :p