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We are given that the matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) has a determinant of 2.

We need to find the determinant of the matrix:

\[
\begin{bmatrix}
a & b & c \\
g - 5a & h - 5b & i - 5c \\
2d + 4a & 2e + 4b & 2f + 4c
\end{bmatrix}
\]

### Step-by-Step Approach:

1. **Row operations and determinant properties:**
   Using the fact that determinant calculations are affected by row operations, we notice that the second row is a linear transformation of the first row, and the third row is also a combination of the original second row and first row.

2. **Row manipulation and factorization:**
   Let us factor out the constants from rows two and three:
   - The second row has a factor of \( -5 \) multiplied by the first row.
   - The third row has a factor of \( 4 \) from the first row and a factor of \( 2 \) from the second row.

3. **Using the determinant properties:**
   The determinant of a matrix where rows are linear combinations of other rows can be expressed as a product of these factors. We use this to simplify our calculation.

Now, I will compute the determinant based on these transformations.The determinant of the given matrix is:

\[
\det\left( \begin{bmatrix} a & b & c \\ g - 5a & h - 5b & i - 5c \\ 2d + 4a & 2e + 4b & 2f + 4c \end{bmatrix} \right) = 2(aei - afh - bdi + bfg + cdh - ceg)
\]

This is exactly twice the determinant of the original matrix \( A \), and since the determinant of \( A \) is given to be 2, we conclude that:

\[
\det\left( \begin{bmatrix} a & b & c \\ g - 5a & h - 5b & i - 5c \\ 2d + 4a & 2e + 4b & 2f + 4c \end{bmatrix} \right) = 2 \times 2 = 4
\]

Thus, the determinant of the transformed matrix is 4.

Do you have any further questions or would you like a deeper explanation of the steps?

Here are some additional questions that may expand on this concept:
1. How do row operations affect the determinant of a matrix?
2. Why does factoring out constants from rows affect the determinant proportionally?
3. What is the geometric interpretation of the determinant in linear transformations?
4. How would the result change if we had different row combinations in the matrix?
5. Can the determinant of a matrix be zero even if none of its entries are zero?

**Tip**: Always remember that multiplying a row by a scalar multiplies the determinant by that scalar.

Let the matrix A = [ [a, b, c], [d, e, f], [g, h, i] ] have determinant 2. What is the determinant of the matrix [ [a, b, c], [g - 5a, h - 5b, i - 5c], [2d + 4a, 2e + 4b, 2f + 4c] ]?

Find the determinant of a matrix after applying row operations and transformations. This problem focuses on linear algebra concepts such as matrix determinants and row operations, suitable for university-level students.

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