Introduction:
Baji Rao I, the eldest son of Balaji Vishwanath, played a crucial role in transforming the Maratha state from the kingdom of Maharashtra into an empire expanding in the north. His military and administrative prowess, along with his strategic alliances, significantly expanded the Maratha influence across northern India.

Expansion of Maratha Empire:
During his tenure as the Peshwa, Baji Rao I undertook several military campaigns and diplomatic maneuvers that led to the expansion of the Maratha Empire.

Military Campaigns:
1. Malwa Campaign: Baji Rao I initiated a series of military campaigns in Malwa, defeating the Mughal governor and establishing Maratha control in the region.
2. Gujarat Campaign: He led successful expeditions in Gujarat, capturing important cities like Ahmedabad and Broach from the Mughals.
3. Bundelkhand Campaign: Baji Rao I launched a campaign in Bundelkhand, defeating the Bundela Rajputs and establishing Maratha authority in the region.
4. Deccan Campaign: He consolidated Maratha hold over the Deccan region by defeating the Nizam of Hyderabad and the Mughals in several battles.

Diplomatic Alliances:
1. Alliance with the Jats: Baji Rao I formed a strategic alliance with the Jats of Bharatpur, which added to the Maratha military strength and helped in their expansion towards Delhi.
2. Treaty with the Mughals: Baji Rao I signed a treaty with the Mughal emperor Muhammad Shah, securing the title of 'Peshwa' and acquiring vast territories in the north.
3. Treaty with the Nizam: He also entered into a treaty with the Nizam of Hyderabad, ensuring a peaceful coexistence and consolidating Maratha control in the Deccan.

Contribution to Maratha Administration:
1. Efficient Administration: Baji Rao I implemented administrative reforms to streamline governance and improve revenue collection, strengthening the Maratha state.
2. Promotion of Art and Culture: He patronized art, literature, and architecture, contributing to the cultural development of the Maratha Empire.
3. Strengthening the Peshwa Institution: Baji Rao I laid the foundation for a strong Peshwa institution, which would continue to play a significant role in Maratha politics even after his demise.

Conclusion:
Through his military campaigns and strategic alliances, Baji Rao I transformed the Maratha state into a powerful empire, expanding its influence in the north. His administrative reforms and patronage of art and culture further contributed to the growth and development of the Maratha Empire.