What is the ruling on authorizing the bank to invest a sum of money in a "Murabaha deposit" where the bank purchases a commodity and sells it on my behalf for a pre-determined profit, and is Zakat obligatory on this amount?

Perhaps the intended meaning of "Murabaha deposit" is what is known as reverse Murabaha or Murabaha-based investment. The Islamic Fiqh Academy has issued a ruling prohibiting this transaction because it is similar to the legally forbidden 'Inah (buy-back) transaction, falls under the concept of forbidden organized Tawarruq (monetization), and contradicts the goal of Islamic finance.

As for the Zakat (definition) on an investment deposit, it is due on both its principal and profit every year, upon the completion of a full year from the day its principal, which constitutes the Nisab (definition), was acquired.