{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T02:41:09Z","timestamp":1780368069367,"version":"3.54.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T00:00:00Z","timestamp":1775606400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,4,8]],"date-time":"2026-04-08T00:00:00Z","timestamp":1775606400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2026,5]]},"DOI":"10.1007\/s12190-026-02813-z","type":"journal-article","created":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T13:34:45Z","timestamp":1777469685000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Some formulas on the Drazin inverse for additive results of matrices and modified matrix"],"prefix":"10.1007","volume":"72","author":[{"given":"Daochang","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7431-7498","authenticated-orcid":false,"given":"Yue","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dijana","family":"Mosi\u0107","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,8]]},"reference":[{"issue":"8","key":"2813_CR1","doi-asserted-by":"publisher","first-page":"863","DOI":"10.1080\/03081087.2010.512291","volume":"59","author":"C. Bu","year":"2011","unstructured":"Bu, C., Zhang, K., Zhao, J.: Representations of the Drazin inverse on solution of a class singular differential equations. Linear Multilinear Algebra 59(8), 863\u2013877 (2011). https:\/\/doi.org\/10.1080\/03081087.2010.512291","journal-title":"Linear Multilinear Algebra"},{"issue":"14","key":"2813_CR2","doi-asserted-by":"publisher","first-page":"7632","DOI":"10.1016\/j.amc.2013.01.050","volume":"219","author":"I. Kyrchei","year":"2013","unstructured":"Kyrchei, I.: Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations. Appl. Math. Comput. 219(14), 7632\u20137644 (2013). https:\/\/doi.org\/10.1016\/j.amc.2013.01.050","journal-title":"Appl. Math. Comput."},{"key":"2813_CR3","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1016\/j.amc.2014.03.125","volume":"238","author":"I. Kyrchei","year":"2014","unstructured":"Kyrchei, I.: Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations. Appl. Math. Comput. 238, 193\u2013207 (2014). https:\/\/doi.org\/10.1016\/j.amc.2014.03.125","journal-title":"Appl. Math. Comput."},{"issue":"3","key":"2813_CR4","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1137\/1017044","volume":"17","author":"C.D. Meyer, Jr.","year":"1975","unstructured":"Meyer, Jr., C.D.: The role of the group generalized inverse in the theory of finite Markov chains. SIAM Rev. 17(3), 443\u2013464 (1975). https:\/\/doi.org\/10.1137\/1017044","journal-title":"SIAM Rev."},{"issue":"4","key":"2813_CR5","doi-asserted-by":"publisher","first-page":"699","DOI":"10.1137\/0714047","volume":"14","author":"C.D. Meyer, Jr.","year":"1977","unstructured":"Meyer, Jr., C.D., Plemmons, R.J.: Convergent powers of a matrix with applications to iterative methods for singular Linear systems. SIAM J. Numer. Anal. 14(4), 699\u2013705 (1977). https:\/\/doi.org\/10.1137\/0714047","journal-title":"SIAM J. Numer. Anal."},{"key":"2813_CR6","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1016\/j.cam.2016.01.059","volume":"301","author":"J.R. Sendra","year":"2016","unstructured":"Sendra, J.R., Sendra, J.: Symbolic computation of Drazin inverses by specializations. J. Comput. Anal. Appl. 301, 201\u2013212 (2016). https:\/\/doi.org\/10.1016\/j.cam.2016.01.059","journal-title":"J. Comput. Anal. Appl."},{"key":"2813_CR7","doi-asserted-by":"publisher","unstructured":"Behera, R., Nandi, A.K., Sahoo, J.K.: Further results on the Drazin inverse of even-order tensors. Numer. Linear Algebra App. 27(5), e2317 (2020). https:\/\/doi.org\/10.1002\/nla.2317","DOI":"10.1002\/nla.2317"},{"issue":"4","key":"2813_CR8","doi-asserted-by":"publisher","first-page":"853","DOI":"10.2298\/FIL1504853Z","volume":"29","author":"D. Zhang","year":"2015","unstructured":"Zhang, D., Du, X.: Representations for the Drazin inverse of a modified matrix. Filomat 29(4), 853\u2013863 (2015). https:\/\/doi.org\/10.2298\/FIL1504853Z","journal-title":"Filomat"},{"key":"2813_CR9","doi-asserted-by":"publisher","first-page":"129776","DOI":"10.1016\/j.amc.2025.129776","volume":"512","author":"D. Zhang","year":"2026","unstructured":"Zhang, D., Ma, L., Mosi\u0107, D.: Representations of the Moore-Penrose inverse of a 2\u00d72 block matrix based on Schur complements. Appl. Math. Comput. 512, 129776 (2026). https:\/\/doi.org\/10.1016\/j.amc.2025.129776","journal-title":"Appl. Math. Comput."},{"key":"2813_CR10","doi-asserted-by":"publisher","first-page":"482","DOI":"10.1016\/j.laa.2019.07.033","volume":"582","author":"D. Zhang","year":"2019","unstructured":"Zhang, D., Mosi\u0107, D., Tam, T.: On the existence of group inverses of Peirce corner matrices. Linear Algebra Appl. 582, 482\u2013498 (2019). https:\/\/doi.org\/10.1016\/j.laa.2019.07.033","journal-title":"Linear Algebra Appl."},{"key":"2813_CR11","doi-asserted-by":"publisher","first-page":"115187","DOI":"10.1016\/j.cam.2023.115187","volume":"428","author":"D. Zhang","year":"2023","unstructured":"Zhang, D., Zhao, Y., Mosi\u0107, D., Katsikis, V.N.: Exact expressions for the Drazin inverse of anti-triangular matrices. J. Comput. Appl. Math. 428, 115187 (2023). https:\/\/doi.org\/10.1016\/j.cam.2023.115187","journal-title":"J. Comput. Appl. Math."},{"key":"2813_CR12","doi-asserted-by":"publisher","first-page":"116701","DOI":"10.1016\/j.cam.2025.116701","volume":"470","author":"D. Zhang","year":"2025","unstructured":"Zhang, D., Zhao, Y., Mosi\u0107, D.: The generalized Drazin inverse of the sum of two elements in a banach algebra. J. Comput. Appl. Math. 470, 116701 (2025). https:\/\/doi.org\/10.1016\/j.cam.2025.116701","journal-title":"J. Comput. Appl. Math."},{"issue":"7","key":"2813_CR13","doi-asserted-by":"publisher","first-page":"506","DOI":"10.1080\/00029890.1958.11991949","volume":"65","author":"M.P. Drazin","year":"1958","unstructured":"Drazin, M.P.: Pseudo-inverses in associative rings and semigroups. Am. Math. 65(7), 506\u2013514 (1958). https:\/\/doi.org\/10.1080\/00029890.1958.11991949","journal-title":"Am. Math."},{"issue":"1","key":"2813_CR14","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1016\/j.jmaa.2008.09.035","volume":"350","author":"N. Castro-Gonz\u00e1lez","year":"2008","unstructured":"Castro-Gonz\u00e1lez, N., Dopazo, E., Mart\u00ednez-Serrano, M.F.: On the Drazin inverse of the sum of two operators and its application to operator matrices. J. Math. Anal. Appl. 350(1), 207\u2013215 (2008). https:\/\/doi.org\/10.1016\/j.jmaa.2008.09.035","journal-title":"J. Math. Anal. Appl."},{"issue":"2","key":"2813_CR15","doi-asserted-by":"publisher","first-page":"351","DOI":"10.1007\/s41980-020-00521-3","volume":"48","author":"A. Shakoor","year":"2022","unstructured":"Shakoor, A., Ali, I., Wali, S., Rehman, A.: Some formulas on the Drazin inverse for the sum of two matrices and block matrices. Bull. Iran Math. Soc. 48(2), 351\u2013366 (2022). https:\/\/doi.org\/10.1007\/s41980-020-00521-3","journal-title":"Bull. Iran Math. Soc."},{"issue":"20","key":"2813_CR16","doi-asserted-by":"publisher","first-page":"10226","DOI":"10.1016\/j.amc.2012.03.102","volume":"218","author":"C. Bu","year":"2012","unstructured":"Bu, C., Feng, C., Bai, S.: Representations for the Drazin inverses of the sum of two matrices and some block matrices. Appl. Math. Comput. 218(20), 10226\u201310237 (2012). https:\/\/doi.org\/10.1016\/j.amc.2012.03.102","journal-title":"Appl. Math. Comput."},{"issue":"12","key":"2813_CR17","doi-asserted-by":"publisher","first-page":"3683","DOI":"10.1016\/j.cam.2011.02.029","volume":"235","author":"J. Ljubisavljevi\u0107","year":"2011","unstructured":"Ljubisavljevi\u0107, J., Cvetkovi\u0107-Ili\u0107, D.S.: Additive results for the Drazin inverse of block matrices and applications. J. Comput. Appl. Math. 235(12), 3683\u20133690 (2011). https:\/\/doi.org\/10.1016\/j.cam.2011.02.029","journal-title":"J. Comput. Appl. Math."},{"issue":"2","key":"2813_CR18","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1137\/1031049","volume":"31","author":"W.W. Hager","year":"1989","unstructured":"Hager, W.W.: Updating the inverse of a matrix. SIAM Rev. 31(2), 221\u2013239 (1989). https:\/\/doi.org\/10.1137\/1031049","journal-title":"SIAM Rev."},{"issue":"2\u20133","key":"2813_CR19","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1016\/S0096-3003(00)00131-4","volume":"125","author":"Y. Wei","year":"2002","unstructured":"Wei, Y.: The Drazin inverse of a modified matrix. Appl. Math. Comput. 125(2\u20133), 295\u2013301 (2002). https:\/\/doi.org\/10.1016\/S0096-3003(00)00131-4","journal-title":"Appl. Math. Comput."},{"issue":"4","key":"2813_CR20","doi-asserted-by":"publisher","first-page":"1678","DOI":"10.1016\/j.laa.2011.06.023","volume":"438","author":"E. Dopazo","year":"2013","unstructured":"Dopazo, E., Mart\u00ednez-Serrano, M.F.: On deriving the Drazin inverse of a modified matrix. Linear Algebra Appl. 438(4), 1678\u20131687 (2013). https:\/\/doi.org\/10.1016\/j.laa.2011.06.023","journal-title":"Linear Algebra Appl."},{"issue":"2","key":"2813_CR21","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1016\/S0252-9602(12)60031-9","volume":"32","author":"D.S. Cvetkovi\u0107-Ili\u0107","year":"2012","unstructured":"Cvetkovi\u0107-Ili\u0107, D.S., Ljubisavljevi\u0107, J.: A note on the Drazin inverse of a modified matrix. Acta Math. Sci. Ser, B 32(2), 483\u2013487 (2012). https:\/\/doi.org\/10.1016\/S0252-9602(12)60031-9","journal-title":"Acta Math. Sci. Ser, B"},{"issue":"4","key":"2813_CR22","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1007\/s10092-012-0069-x","volume":"50","author":"D. Mosi\u0107","year":"2013","unstructured":"Mosi\u0107, D.: Some results on the Drazin inverse of a modified matrix. Calcolo 50(4), 305\u2013311 (2013). https:\/\/doi.org\/10.1007\/s10092-012-0069-x","journal-title":"Calcolo"},{"issue":"3","key":"2813_CR23","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1007\/s10092-013-0098-0","volume":"51","author":"A. Shakoor","year":"2014","unstructured":"Shakoor, A., Yang, H., Ali, I.: Some representations for the Drazin inverse of a modified matrix. Calcolo 51(3), 505\u2013514 (2014). https:\/\/doi.org\/10.1007\/s10092-013-0098-0","journal-title":"Calcolo"},{"key":"2813_CR24","doi-asserted-by":"publisher","unstructured":"Zhao, Y., Zhang, D., Mosi\u0107, D.: Formulae for the Drazin inverse of modified tensors via the Einstein product. https:\/\/doi.org\/10.48550\/arXiv.2604.21818","DOI":"10.48550\/arXiv.2604.21818"},{"key":"2813_CR25","unstructured":"Cline, R.E.: An application of representation for the generalized inverse of a matrix. MRC Technical Report 592 (1965)"},{"issue":"1\u20133","key":"2813_CR26","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1016\/S0024-3795(00)00257-3","volume":"322","author":"R.E. Hartwig","year":"2001","unstructured":"Hartwig, R.E., Wang, G., Wei, Y.: Some additive results on Drazin inverse. Linear Algebra Appl. 322(1\u20133), 207\u2013217 (2001). https:\/\/doi.org\/10.1016\/S0024-3795(00)00257-3","journal-title":"Linear Algebra Appl."},{"issue":"1","key":"2813_CR27","doi-asserted-by":"publisher","first-page":"10","DOI":"10.1017\/S1446788700020036","volume":"24","author":"R.E. Hartwig","year":"1977","unstructured":"Hartwig, R.E., Shoaf, J.M.: Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices. J. Aust. Math. Soc. 24(1), 10\u201334 (1977). https:\/\/doi.org\/10.1017\/S1446788700020036","journal-title":"J. Aust. Math. Soc."},{"issue":"1","key":"2813_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/0133001","volume":"33","author":"C.D. Meyer, Jr.","year":"1977","unstructured":"Meyer, Jr., C.D., Rose, N.J.: The index and the Drazin inverse of block triangular matrices. SIAM J. Appl. Math. 33(1), 1\u20137 (1977). https:\/\/doi.org\/10.1137\/0133001","journal-title":"SIAM J. Appl. Math."},{"issue":"4","key":"2813_CR29","doi-asserted-by":"publisher","first-page":"2699","DOI":"10.1007\/s12190-021-01638-2","volume":"68","author":"D. Zhang","year":"2022","unstructured":"Zhang, D., Jin, Y., Mosi\u0107, D.: The Drazin inverse of anti-triangular block matrices. J. Appl. Math. Comput. 68(4), 2699\u20132716 (2022). https:\/\/doi.org\/10.1007\/s12190-021-01638-2","journal-title":"J. Appl. Math. Comput."},{"key":"2813_CR30","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1016\/j.laa.2004.12.027","volume":"400","author":"N. Castro-Gonz\u00e1lez","year":"2005","unstructured":"Castro-Gonz\u00e1lez, N., Dopazo, E.: Representations of the Drazin inverse for a class of block matrices. Linear Algebra Appl. 400, 253\u2013269 (2005). https:\/\/doi.org\/10.1016\/j.laa.2004.12.027","journal-title":"Linear Algebra Appl."},{"issue":"18","key":"2813_CR31","doi-asserted-by":"publisher","first-page":"6387","DOI":"10.2298\/FIL1818387S","volume":"32","author":"L. Stojkovi\u0107","year":"2018","unstructured":"Stojkovi\u0107, L., Mosi\u0107, D.: Expressions for the Drazin inverse of a modified matrix. Filomat 32(18), 6387\u20136394 (2018). https:\/\/doi.org\/10.2298\/FIL1818387S","journal-title":"Filomat"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02813-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-026-02813-z","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02813-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T12:30:17Z","timestamp":1779885017000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-026-02813-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,8]]},"references-count":31,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2026,5]]}},"alternative-id":["2813"],"URL":"https:\/\/doi.org\/10.1007\/s12190-026-02813-z","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,8]]},"assertion":[{"value":"24 February 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 April 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 April 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 April 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"164"}}