Geodesic
On cofactor expansion of determinants of Boolean matrices
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 4, pp. 199-223.

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We give necessary and sufficient conditions of a cofactor expansibility of determinants along a row or column for Boolean square matrices over an arbitrary Boolean algebra. First of all we define a natural decomposition of an arbitrary Boolean matrix by interior, exterior, and determinate parts. The introduced notions allow us to establish the main result of this paper. It is shown that the formulas of the cofactor expansion along a row (column) of determinants of an arbitrary square Boolean matrix hold if and only if the formulas of the cofactor expansion along the corresponding row (column) hold for determinants of its interior part. 
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V. B. Poplavskii. On cofactor expansion of determinants of Boolean matrices. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 4, pp. 199-223. http://geodesic.mathdoc.fr/item/FPM_2007_13_4_a9/

[8] Poplavskii V. B., “Obratimye i prisoedinennye bulevy matritsy”, Chebyshevskii sb., 6:1 (2005), 174–181 | MR | Zbl

[9] Skornyakov L. A., “Obratimye matritsy nad distributivnymi strukturami”, Sib. mat. zhurn., 27:2 (1986), 182–185 | MR | Zbl

[10] Chesley D. S., Bevis J. H., “Determinants for matrices over lattices”, Proc. Roy. Soc. Edinburgh Sect. A, 68:2 (1969), 138–144 | MR | Zbl

[11] Golan J. S., Semirings and Their Applications, Kluwer Academic, Dordrecht, 1999 | MR

[12] Poplavski V., “On orientability and degeneration of Boolean binary relation on a finite set”, Mathematical Logic in Asia, Proc. 9th Asian Logic Conf. (Novosibirsk, Russia, 16–19 August 2005), World Scientific, Singapore, 2006, 203–214 | MR | Zbl

[13] Poplin P. L., Hartwig R. E., “Determinantal identities over commutative semirings”, Linear Algebra Appl., 387 (2004), 99–132 | DOI | MR | Zbl

[14] Reutenauer C., Straubing H., “Inversion of matrices over a commutative semiring”, J. Algebra, 88 (1984), 350–360 | DOI | MR | Zbl

[15] Rutherford D. E., “Inverses of Boolean matrices”, Proc. Glasgow Math. Assoc., 6:1 (1963), 49–53 | DOI | MR | Zbl