= Solution
Let $P$ have columns $a,b,c$ and let $G$ have columns $A,B,C$. The cross-product formulas defining the <reciprocal basis> give
$$
G^TP=I,\qquad G=P^{-T},
$$
provided the original <scalar triple product> $T=\det P$ is nonzero. Taking <determinants> yields the <reciprocal basis involution> identity
$$
\boxed{\langle A,B,C\rangle=\det G=\frac1{\det P}=\frac1{\langle a,b,c\rangle}.}
$$
When $T=0$, the displayed reciprocal vectors are undefined, so there is no additional zero-denominator case to evaluate.
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