All the proposed Hill cipher modifications have been restricted to the use of dynamic keys only. In this paper, we propose an extension of Hill cipher modification based on eigenvalues HCM-EE, called HCM-EXDKS. The proposed extension generating dynamic encryption key matrix by exponentiation that is made efficiently with the help of eigenvalues, HCM-EXDKS introduces a new class of dynamic keys together with dynamically changing key size. Security of HCM-EXDKS is provided by the use of a large number of dynamic keys with variable size. The proposed extension is more effective in the encryption quality of RGB images than HCM-EE and Hill cipher-known modifications in the case of images with large single colour areas and slightly more effective otherwise. HCM-EXDKS almost has the same encryption time as HCM-EE, and HCM-HMAC. HCM-EXDKS is two times faster than HCM-H, having the best encryption quality among Hill cipher modifications compared versus HCM-EXDKS.