To calculate the determinant of matrix $A$, we need to consider the sequence of operations performed during Gaussian elimination. Gaussian elimination involves row operations, which can affect the determinant as follows:

1. Row Swapping: Changes the sign of the determinant.
2. Row Multiplication by a Scalar: Multiplies the determinant by that scalar.
3. Adding/Subtracting a Multiple of One Row to Another: Does not change the determinant.

Here are the steps based on the Gaussian elimination process provided:

• The final matrix after all transformations is -3 & -9 & 3 \\ 0 & 8 & 3 \\ 0 & 0 & -1 \end{bmatrix}$$

Determinant of the Final Matrix:

The determinant of an upper triangular matrix is the product of its diagonal elements: $\text{det}(A) = (-3) \cdot (8) \cdot (-1) = 24.$

Verifying Determinant Adjustments:

• None of the row operations in the elimination steps change the determinant’s value significantly (row scaling or swapping are not explicitly mentioned).

Thus, the determinant of $A$ is:

$\boxed{24}$

Let me know if you want further details or explanations!

Here are five related questions to expand your understanding:

1. How do elementary row operations impact a matrix's determinant?
2. Why is the determinant of an upper triangular matrix simply the product of its diagonal entries?
3. Can the determinant be used to test if a matrix is invertible? Why or why not?
4. How does the determinant relate to eigenvalues of a matrix?
5. What are the real-life applications of determinants in physics or engineering?

Tip: Always ensure that you track the row operations and their effects on the determinant when performing Gaussian elimination.