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(A^k^"^1)),XB=D(N(X)@?N(B^k^"^2)),AXB=D(R(X)@?R(A^k^"^1),N(X)@?N(B^k^"^2)),where A@?C^n^x^n with Ind(A)=k"1, B@?C^m^x^m with Ind(B)=k"2, and D@?C^n^x^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.