To determine the inverse of 9 in this operation table with 6 as the identity element, we should look for a value that, when combined with 9, results in 6 in both directions (i.e., in the rows and columns).

Here's a step-by-step analysis:

1. Identify the identity element (6):

   • The identity element 6 should satisfy the property that any element combined with it returns that element itself.
   • For example, if we combine 6 with any element $x$, it should give $x$ back.

2. Find the inverse of 9:

   • The inverse of 9 is the element that, when combined with 9, yields 6 in the table.
   • We need to check the row and column corresponding to 9 to see if any combination with another number gives 6.

Looking at the operation table provided:

• In the row for 9 (last row), combining 9 with 7 yields 6.
• Similarly, in the column for 9 (last column), combining 9 with 7 also yields 6.

Thus, the inverse of 9 in this operation table is 7, because $9 \ast 7 = 6$ and $7 \ast 9 = 6$.

Would you like further clarification or additional details? Here are some related questions to deepen understanding:

1. What is the role of an identity element in an operation table?
2. How do we verify the identity element in a given table?
3. Can an operation table have more than one identity element?
4. What properties make an element the inverse of another in a group?
5. Are there cases where an element might not have an inverse?

Tip: When dealing with operation tables, check both row and column intersections to confirm properties like identity and inverse.