Geodesic
Some structural properties of inverse matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part VII, Tome 139 (1984), pp. 61-73

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The connection of some structural properties of inverse matrices such as the presence of an invertible submatrix or of a rectangular submatrix of incomplete rank is investigated. Explicit relations are obtained between the ranks of dual submatrices. The results are applied to describe the inverse to generalized Hessenberg and band matrices. 
@article{ZNSL_1984_139_a4,
     author = {L. Yu. Kolotilina},
     title = {Some structural properties of inverse matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {61--73},
     publisher = {mathdoc},
     volume = {139},
     year = {1984},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_139_a4/}
}
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L. Yu. Kolotilina. Some structural properties of inverse matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part VII, Tome 139 (1984), pp. 61-73. http://geodesic.mathdoc.fr/item/ZNSL_1984_139_a4/