The solvability of two linear matrix equations

The solvability conditions of the following two linear matrix equations (i) A(1)X(1)B(1) + A(2)X(2)B(2) + A(3)X(3)B(3) = C, (ii) A(1)XB(1) = C-1, A(2)XB(2) = C-2, A(3)XB(3) = C-3 are established using ranks and generalized inverses of matrices. In additi on, the duality of the three types of matrix equations (iii) A(1)X(1)B(1) +A(2)X(2)B(2) +A(3)X(3)B(3) + A(4)X(4)B(4) = C, (iv) A(1)XB(1) = C-1, A(2)XB(2) = C-2, A(3)XB(3) = C-3, A(4)XB(4) = C-4, (v) AXB+CXD=E are also considered.