Solving a kind of restricted matrix equations and Cramer rule

Abstract

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 equations AX=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× n with Ind( A)= k 1, B∈ C m× m with Ind( B)= k 2, and D∈ C n× 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.