How does the concept of mutual inductance contribute to the design and operation of transformers in electrical engineering? Discuss its role in efficiently transferring energy between primary and secondary coils, the significance of core materials in enhancing magnetic coupling, and the factors affecting mutual inductance in practical transformer designs. Additionally, analyze the impact of mutual inductance on transformer efficiency and explore methods to mitigate losses associated with leakage flux and eddy currents. Finally, discuss the applications of transformers in various electrical systems, such as power distribution, voltage regulation, and impedance matching, highlighting the importance of mutual inductance in these applications.