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Ishii and Oshiro introduced the notion of an $f$-twisted Alexander matrix,which is a quandle version of a twisted Alexander matrix and defined aninvariant of finitely presented quandles. In this paper, we study $f$-twistedAlexander matrices of certain quandles with the Alexander pair obtained from aquandle 2-cocycle. We show that the 0-th elementary ideal of $f$-twistedAlexander matrix of the knot quandle of a surface knot with the Alexander pairobtained from a quandle 2-cocycle can be described with theCarter-Saito-Satoh's invariant. We also discuss a relationship between$f$-twisted Alexander matrices of connected quandles with the Alexander pairobtained from a quandle 2-cocycle and quandle homology groups.

Twisted Alexander matrices of quandles associated with a certain Alexander pair