We are tasked with finding the determinant of the matrix $\begin{pmatrix} a & 6 & d \\ b & 9 & e \\ c & 12 & f \end{pmatrix}$, given the following information:

1. $\det\begin{pmatrix} a & 1 & d \\ b & 2 & e \\ c & 3 & f \end{pmatrix} = -4$
2. $\det\begin{pmatrix} a & 1 & d \\ b & 1 & e \\ c & 1 & f \end{pmatrix} = 4$

Step-by-Step Approach:

The key to solving this problem lies in understanding how scaling a column of a matrix affects its determinant.

Determinant Properties:

1. Scaling a column: If any column of a matrix is multiplied by a constant $k$, the determinant of the matrix is multiplied by $k$.

Let's examine how the given matrix $\begin{pmatrix} a & 6 & d \\ b & 9 & e \\ c & 12 & f \end{pmatrix}$ relates to the first matrix $\begin{pmatrix} a & 1 & d \\ b & 2 & e \\ c & 3 & f \end{pmatrix}$.

Matrix Structure Comparison:

We observe that the second column in the desired matrix is $\begin{pmatrix} 6 \\ 9 \\ 12 \end{pmatrix}$, which is exactly 3 times the second column of the matrix $\begin{pmatrix} a & 1 & d \\ b & 2 & e \\ c & 3 & f \end{pmatrix}$, i.e., $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$.

Thus, using the property of scaling a column, the determinant of the matrix $\begin{pmatrix} a & 6 & d \\ b & 9 & e \\ c & 12 & f \end{pmatrix}$ will be 3 times the determinant of the matrix $\begin{pmatrix} a & 1 & d \\ b & 2 & e \\ c & 3 & f \end{pmatrix}$.

Calculation:

$\det\begin{pmatrix} a & 6 & d \\ b & 9 & e \\ c & 12 & f \end{pmatrix} = 3 \times \det\begin{pmatrix} a & 1 & d \\ b & 2 & e \\ c & 3 & f \end{pmatrix}$

Substitute the known determinant:

$\det\begin{pmatrix} a & 6 & d \\ b & 9 & e \\ c & 12 & f \end{pmatrix} = 3 \times (-4) = -12$

Final Answer:

$\det\begin{pmatrix} a & 6 & d \\ b & 9 & e \\ c & 12 & f \end{pmatrix} = -12$

---

Would you like further details or explanations?

Here are some related questions for exploration:

1. What happens to the determinant when two rows of a matrix are swapped?
2. How does the determinant change if all entries in a column are multiplied by different values?
3. How is the determinant affected if a row or column is replaced by a linear combination of other rows/columns?
4. What is the geometric interpretation of the determinant of a 3x3 matrix?
5. Can the determinant of a matrix be zero even if no entries are zero?

Tip: Always remember that the determinant of a matrix is zero if two rows or columns are identical or linearly dependent.