Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 591256, 14 pages
http://dx.doi.org/10.1155/2012/591256
Research Article

An Analog of the Adjugate Matrix for the Outer Inverse 𝐴 ( 2 ) 𝑇 , 𝑆

Xiaoji Liu,1,2 Guangyan Zhu,1 Guangping Zhou,1 and Yaoming Yu3

1School of Science, Guangxi University for Nationalities, Nanning 530006, China
2Guangxi Key Laboratory of Hybrid Computational and IC Design Analysis, Nanning 530006, China
3School of Mathematical Sciences, Monash University, Caulfield East, VIC 3800, Australia

Received 5 August 2011; Revised 24 November 2011; Accepted 7 December 2011

Academic Editor: Kui Fu Chen

Copyright © 2012 Xiaoji Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the determinantal representation by exploiting the limiting expression for the generalized inverse 𝐴 ( 2 ) 𝑇 , 𝑆 . We show the equivalent relationship between the existence and limiting expression of 𝐴 ( 2 ) 𝑇 , 𝑆 and some limiting processes of matrices and deduce the new determinantal representations of 𝐴 ( 2 ) 𝑇 , 𝑆 , based on some analog of the classical adjoint matrix. Using the analog of the classical adjoint matrix, we present Cramer rules for the restricted matrix equation 𝑆 𝐴 𝑋 𝐵 = 𝐷 , ℛ ( 𝑋 ) ⊂ 𝑇 , 𝒩 ( 𝑋 ) ⊃ .