The behaviour of the eigenvalues of Dirichlet Boundary-Domain Integro-Differential Equations (BDIDEs) with a reduced number of collocation points has never been discussed theoretically. The uncertainty of the behaviour of the eigenvalues of Dirichlet BDIDEs will prohibit the use of iteration methods in solving the BDIDEs system of equations. The purpose of this paper is to demonstrate the spectral properties of matrix operator obtained from the discretized Dirichlet BDIDEs with reduced number of collocation points. We calculate the eigenvalues of the matrix operator, numerically. The discussions of the spectral properties are based on the eigenvalues of the discretized BDIDEs that are obtained numerically. In our numerical test, the attribution of the eigenvalues for matrix operator obtained numerically for the discretized BDIDEs with reduced number of collocation points exceeds 1. The findings demonstrate that it is utterly impracticable to solve the system yielded from the discretized Dirichlet BDIDEs with a reduced number of collocation points with an iterative method. The theoretical explanation of why this behaviour occurs is also provided. With this result of the eigenvalues attained, the matrix equations yielded from the discretized BDIDEs with a reduced number of collocation points can purely be solved by direct methods.