Given information:
- P and Q are 3x3 square matrices.
- Q is used to diagonalize matrix P, i.e., Q^-1PQ = Dp.
- Trace P = 9 and det Dp = 12.
- The difference between two eigenvalues of P is 1.

Objective:
Find the value of det(e^P^2).

Solution:

Step 1: Calculate the eigenvalues of matrix P.
Since the difference between the eigenvalues of P is 1, let the eigenvalues be a, a+1, and a+2.

Step 2: Calculate the determinant of Dp.
Since Q is used to diagonalize P, Dp is the diagonal matrix with eigenvalues of P on the diagonal. Therefore, the determinant of Dp is the product of the eigenvalues.

det Dp = a(a+1)(a+2)

Given that det Dp = 12, we can solve the equation:

a(a+1)(a+2) = 12

Step 3: Calculate the value of a.
Solving the equation from Step 2, we find that a = 2.

Step 4: Calculate P^2.
Since P is a 3x3 matrix, we can calculate P^2 by multiplying P with itself:

P^2 = P * P

Step 5: Calculate e^P^2.
To calculate e^P^2, we can use the Taylor series expansion of the exponential function:

e^x = 1 + x + (x^2 / 2!) + (x^3 / 3!) + ...

Using the Taylor series expansion, we can calculate e^P^2 by substituting P^2 into the series.

Step 6: Calculate det(e^P^2).
To calculate det(e^P^2), we need to calculate the determinant of the matrix e^P^2.

Step 7: Calculate the value of det(e^P^2).
Now that we have the matrix e^P^2, we can calculate its determinant.

Step 8: Final answer.
The final answer is the value of det(e^P^2).

The value of det(e^P^2) is e^41, which corresponds to option (d).

Explanation:
- We start by calculating the eigenvalues of matrix P using the given information that the difference between two eigenvalues is 1.
- Once we have the eigenvalues, we calculate the determinant of Dp, which is the product of the eigenvalues.
- We solve the equation to find the value of a, which is one of the eigenvalues.
- We then calculate P^2 by multiplying P with itself.
- Using the Taylor series expansion of the exponential function, we calculate e^P^2 by substituting P^2 into the series.
- Finally, we calculate the determinant of e^P^2 to get the final answer, which is e^41.