This problem involves properties of determinants.

Key Property of Determinants:

• Adding a multiple of one row to another row in a matrix does not change the determinant.

Solution:

The matrix $D$ is obtained from $A$ by adding $6$ times the fourth row to the second row. According to the above determinant property, such a row operation leaves the determinant unchanged.

Therefore, $\det(D) = \det(A) = 3.$

Final Answer:

$\det(D) = 3$

Let me know if you would like a further explanation or any related questions! Here are 5 related questions you might explore:

1. What happens to the determinant if two rows of a matrix are swapped?
2. How does multiplying one row of a matrix by a scalar affect the determinant?
3. What determinant properties hold if a matrix is triangular?
4. How is the determinant of a matrix affected when a matrix is transposed?
5. How can row reduction be used to compute the determinant of a matrix?

Tip: Understanding the effect of row operations on determinants can significantly simplify computations.