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In this paper we give necessary and sufficient conditions for the numerical range of a 5 × 5 matrix to be a convex hull of three points and one ellipse or one point and two ellipses. We also find sufficient conditions for which the numerical range of a 5 × 5 matrix is an elliptic disc.

In this paper we obtain numerical radius of a general complex matrix of order two without reducing it in triangular form. We also find numerical radius of nilpotent matrix of order three and different matrices of order three and four satisfying special property.