Department of Mathematics
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The applications of generalized inverses of matrices appear in many fields like mathematical finance, applied mathematics and engineering. In this talk, we discuss generalized inverses of matrices over a skew polynomial ring \( S = R[x;\sigma,\delta] \), where \( R \) is a division ring.

We first introduce some necessary and sufficient conditions for the existence of \( \lbrace 1\rbrace\)-, \( \lbrace 1,2\rbrace \)-, \( \lbrace 1,3\rbrace \)-, \(\lbrace 1,4\rbrace\)- and MP-inverses of Ore matrices, and give some explicit formulas over \( S \). We also explore the solutions of linear systems over \( S \) by using \(\lbrace 1\rbrace\)-inverses of Ore matrices. Next, we extend Roth's Theorem 1 and generalized Roth's Theorem 1 to the Ore matrices case. Furthermore, we consider the extensions of all the involutions \( \psi \) on \( R(x) \), and construct some necessary and sufficient conditions for \( \psi \) to be an involution on \( R(x)[D;\sigma,\delta] \). Finally, we find two different explicit formulas for \( \lbrace 1,3\rbrace \)- and \( \lbrace 1,4\rbrace \)-inverses of Ore matrices.