Condensed cramer rule for solving restricted matrix equations

A Cramer rule for solving a kind of restricted matrix equationsWAWXW~BW~=D,R(X)@?R[(AW)^k^"^1],N(X)@?N[(W~B)^k^"^2^@?]is presented in [G. Wang, Jie Sun, A Cramer rule for solution of the general restricted matrix equation, Appl. Math. Comput. 154 (2004) 415-422]. This paper gives a more condensed Cramer rule for the solution of the matrix equations. The results in [G. Wang, Z.L. Xu, Solving a kind of restricted matrix equations and Cramer rule, Appl. Math. Comput. 162 (2005) 329-338, Chao Gu, G.R. Wang, Z.L. Xu, PCR Algorithm for the Parallel Computation of the Solution of a class of Singular linear Systems, Appl. Math. Comput. 176 (2006) 237-244] are partially the special cases in our paper.