  We revisit the class of column competent matrices and study some matrix
theoretic properties of this class. The local $w$-uniqueness of the solutions
to the linear complementarity problem can be identified by the column competent
matrices. We establish some new results on $w$-uniqueness properties in
connection with column competent matrices. These results are significant in the
context of matrix theory as well as algorithms in operations research. We prove
some results in connection with locally $w$-uniqueness property of column
competent matrices. Finally we establish a connection between column competent
matrices and column adequate matrices with the help of degree theory.
