{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T14:37:53Z","timestamp":1777387073987,"version":"3.51.4"},"reference-count":13,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2006,12,8]],"date-time":"2006-12-08T00:00:00Z","timestamp":1165536000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2007,4]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, first we establish a determinantal representation for the group inverse <jats:italic>A<\/jats:italic><jats:sub><jats:italic>g<\/jats:italic><\/jats:sub> of a square matrix <jats:italic>A<\/jats:italic>. Based on this, a determinantal representation for the generalized inverse <jats:italic>A<\/jats:italic><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/tex2gif-stack-3.gif\" xlink:title=\"urn:x-wiley:10705325:media:NLA513:tex2gif-stack-3\"\/> is presented. As an application, we give a determinantal formula for the unique solution of the general restricted linear system: <jats:italic>Ax<\/jats:italic>=<jats:italic>b<\/jats:italic>(<jats:italic>x \u2208 T<\/jats:italic>, <jats:italic>b \u2208 AT<\/jats:italic> and dim(<jats:italic>AT<\/jats:italic>)=dim(<jats:italic>T<\/jats:italic>)), which reduces to the common Cramer rule if <jats:italic>A<\/jats:italic> is non\u2010singular. These results extend our earlier work. Copyright \u00a9 2006 John Wiley &amp; Sons, Ltd.<\/jats:p>","DOI":"10.1002\/nla.513","type":"journal-article","created":{"date-parts":[[2006,12,8]],"date-time":"2006-12-08T14:20:10Z","timestamp":1165587610000},"page":"169-182","source":"Crossref","is-referenced-by-count":29,"title":["On determinantal representation for the generalized inverse <i>A<\/i> and its applications"],"prefix":"10.1002","volume":"14","author":[{"given":"Jing","family":"Cai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guoliang","family":"Chen","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,12,8]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(82)90255-5"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(82)90117-3"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90395-9"},{"key":"e_1_2_1_5_2","first-page":"1","article-title":"On extensions of Cramer rule","volume":"21","author":"Wang GR","year":"1992","journal-title":"Journal of Shanghai Normal University"},{"issue":"2","key":"e_1_2_1_6_2","first-page":"3","article-title":"A Cramer rule for the unique solution of restricted linear system Ax=b, (x \u2208 T)","volume":"16","author":"Chen YL","year":"1993","journal-title":"Journal of Nanjing Normal University"},{"issue":"1","key":"e_1_2_1_7_2","first-page":"61","article-title":"Explicit expression and Cramer rule for solution of restricted linear equations","volume":"8","author":"Chen YL","year":"1993","journal-title":"Applied Mathematics, A Journal of Chinese Universities"},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1080\/03081088408817600"},{"key":"e_1_2_1_9_2","first-page":"320","article-title":"On the representation of A\n                  +, A and its applications","volume":"24","author":"Cai J","year":"2002","journal-title":"Numerical Mathematics, A Journal of Chinese Universities"},{"key":"e_1_2_1_10_2","volume-title":"A Treatise on the Theory of Determinants","author":"Muir T","year":"1960"},{"key":"e_1_2_1_11_2","volume-title":"Generalized Inverse: Theory and Applications","author":"Ben\u2010Israel A","year":"1974"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(98)00008-1"},{"key":"e_1_2_1_13_2","first-page":"81","article-title":"The Drazin inverse of linear operators","volume":"5","author":"Cai DH","year":"1985","journal-title":"Journal of Mathematics"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0096-3003(96)00180-4"}],"container-title":["Numerical Linear Algebra with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnla.513","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/nla.513","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,13]],"date-time":"2023-10-13T00:19:22Z","timestamp":1697156362000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/nla.513"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,12,8]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2007,4]]}},"alternative-id":["10.1002\/nla.513"],"URL":"https:\/\/doi.org\/10.1002\/nla.513","archive":["Portico"],"relation":{},"ISSN":["1070-5325","1099-1506"],"issn-type":[{"value":"1070-5325","type":"print"},{"value":"1099-1506","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,12,8]]}}}