On Singularity of Bordered Matrix of any Matrix Am×n

Abstract:

      In this paper the sufficient and necessary conditions of the nonsingularity of M are to be given.M=[A B C O] is the bordered matrix of any m x n order matrixA, where O is the zero matrix. The inverse matrix of M is experessed as theblocked fcrm: M^-1=[A₁ B₂ C₃ O₄], where C₁ is n ×m order matrix, C₂ is n×r₁ ordermatrix, C₃ is r₂× m order matrix, C₄ is r₂×r₁ order matrix. Under certain conditions, we prove that C₁ is the generalized inverse matrix of A, which defined as A .