  We introduce the notion of K-theoretic duality for extensions of separable
unital nuclear $C^*$-algebras by using K-homology long exact sequence and
cyclic six term exact sequence for K-theory groups of extensions. We then prove
that the Toeplitz extension $\mathcal{T}_A$ of a Cuntz-Krieger algebra
$\mathcal{O}_A$ is the K-theoretic dual of the Toeplitz extension
$\mathcal{T}_{A^t}$ of the Cuntz-Krieger algebra $\mathcal{O}_{A^t}$ for the
transposed matrix $A^t$ of $A$. A pair of isomorphic Cuntz--Krieger algebras
$\mathcal{O}_A$ and $\mathcal{O}_B$ does not necessarily yield the isomorphic
pair of $\mathcal{O}_{A^t}$ and $\mathcal{O}_{B^t}.$ However, as an
application, we may show that two Toeplitz algebras $\mathcal{T}_A$ and
$\mathcal{T}_B$ are isomorphic as $C^*$-algebras if and only if the Toeplitz
algebras $\mathcal{T}_{A^t}$ and $\mathcal{T}_{B^t}$ of their transposed
matrices are isomorphic.
