In recent years, many studies have been done on the numerical range of matrices and bounded operators. In this presentation, the set of unitarily irreducible complex 3×3 matrices whose numerical range has a flat portion will be explicitly described and it will show that this set is connected also if a numerical range be a 2-dimentional shape with one flat portion on its boundary then it can be restored up to unitary similarity. The seminar may aspires to attract attention of the general mathematical readership to fascinating interplay of linear algebra, geometry and analysis.