Solution for Matrix Equation AX-YB=C

Abstract

The inverse of a matrix A can only exist if A is nonsingular. This is an important theorem in linear algebra, one learned in an introductory course. In recent years, needs have been felt in numerous areas of applied mathematics for some kind of inverse like matrix of a matrix that is singular or even rectangular. To fulfill this need, mathematicians discovered that even if a matrix was not invertible, there is still either a left or right sided inverse of that matrix. The inverse moore Penrose is an inverse matrix type denoted by A⁽¹⁾. The inverse moore penrose is an extension of the inverse matrix concept. complex matrices will be used to find matrix inverses. Matrix m×n field F can write as C_m×n with A⁽¹⁾ g-invers of A, the matrix statisfying the equation AA⁽¹⁾A=A. A necessary and suffient conditions is established for solvability of the matrix equation AX-YB=C. Where matix A,B, and C are giving by equation, we can find the solutions by using Penrose equation existence and construction of -inverse to find matrix X and Y satisfying the equation AX-YB=C. Substitute the matrix  and the matrix  to the equation AX-YB=C so that it is proven that the results of AX-YB are matrix C.