(untitled) (Communications in Mathematical Research, 2021)
We study the constrained system of linear equations Ax=b, x∈R(A^k) for A ∈ C^n×n and b ∈C^n, k=Ind(A). When the system is consistent, it is well known that it has a unique A^Db. If the system is inconsistent, then we seek for the least squares solution of the problem and consider _x R(A^k)\|b-A x\|_2, where \|· \|_2 is the 2-norm. For the inconsistent system with a matrix A of index one, it was proved recently that the solution is A^⊕b using the core inverse A^⊕ of A. For matrices of an arbitrary index and an arbitrary b, we show that the solution of the constrained system can be expressed as A^⊕b where A^⊕ is the core-EP inverse of A. We establish two Cramer's rules for the inconsistent constrained least squares solution and develop several explicit expressions for the core-EP inverse of matrices of an arbitrary index. Using these expressions, two Cramer's rules and one Gaussian elimination method for computing the core-EP inverse of matrices of an arbitrary index are proposed in this paper. We also consider the W-weighted core-EP inverse of a rectangular matrix and apply the weighted core-EP inverse to a more general constrained system of linear equations.