For a finite matrix \(A\) of the form \(\begin{pmatrix} aI& B \\ 0 &C\end{pmatrix}\) is proved that if its numerical range \(W(A)\) is a~circular disc centered at~\(a\), then \(a\) must be an eigenvalue of~\(C\). As a~consequence, the author shows, for any finite matrix~\(A\), that: (a)~if \(\partial W(A)\) contains a~circular disc, then its center is an eigenvalue of~\(A\), its geometric multiplicity being strictly less than its algebraic multiplicity, and~(b) if~\(A\) is similar to a~normal matrix, then \(\partial W(A)\) contains no circular disc.