Solving a kind of restricted matrix equations and Cramer rule

The problem of finding determinantal formula for solutions of some restricted linear systems Ax=b has been discussed in [Linear Multilinear Algebra 15 (1984) 319, Linear Algebra Appl. 116 (1989) 27, Linear Multilinear Algebra 34 (1993) 177, Appl. Math. Comput. 125 (2002) 303]. This paper deals with some more extensive cases of this kind of problem and establishes determinantal formulas for solutions of the restricted matrix equationsAX=D(R(X)⊂R(Ak1)),XB=D(N(X)⊃N(Bk2)),AXB=D(R(X)⊂R(Ak1),N(X)⊃N(Bk2)),where A∈Cn×n with Ind(A)=k1, B∈Cm×m with Ind(B)=k2, and D∈Cn×m. The results in [Linear Multilinear Algebra 15 (1984) 319, Linear Algebra Appl. 74 (1986) 213, Linear Multilinear Algebra 34 (1993) 177, Appl. Math. Comput. 125 (2002) 303] are partially the special cases in our paper. The classic Cramer rule is also a special case of our results.