We are given two 3x3 matrices $A$ and $B$, with $\det(A) = 2$ and $\det(B) = 5$. We need to find $\det \left( (3A)^{-1} B^2 \right)$.

Step 1: Determinant properties

We will use the following properties of determinants:

1. $\det(AB) = \det(A) \cdot \det(B)$ for any square matrices $A$ and $B$.
2. $\det(A^{-1}) = \frac{1}{\det(A)}$ for an invertible matrix $A$.
3. $\det(cA) = c^n \det(A)$ for an $n \times n$ matrix $A$ and scalar $c$, where $n$ is the dimension of the matrix.

Step 2: Determinant of $(3A)^{-1}$

First, calculate the determinant of $3A$: $\det(3A) = 3^3 \det(A) = 27 \cdot 2 = 54$ Now, the determinant of $(3A)^{-1}$ is: $\det \left( (3A)^{-1} \right) = \frac{1}{\det(3A)} = \frac{1}{54}$

Step 3: Determinant of $B^2$

We know that $\det(B^2) = (\det(B))^2$. Therefore: $\det(B^2) = 5^2 = 25$

Step 4: Determinant of the product

Now, use the property $\det(AB) = \det(A) \cdot \det(B)$: $\det \left( (3A)^{-1} B^2 \right) = \det \left( (3A)^{-1} \right) \cdot \det(B^2) = \frac{1}{54} \cdot 25 = \frac{25}{54}$

Thus, the answer is: $\boxed{D. \frac{25}{54}}$

Would you like a deeper explanation of any step?

Here are some related questions to explore:

1. How do you find the inverse of a matrix?
2. What are the properties of matrix multiplication with determinants?
3. How does the determinant change under scalar multiplication?
4. How can determinants help in solving systems of linear equations?
5. What is the geometric interpretation of the determinant of a matrix?

Tip: Understanding matrix determinant properties simplifies many problems involving matrices.