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The problem provides a 3x3 matrix \( S \), and you are tasked with finding its determinant, given that \( |S| = 6 \). Here's a breakdown of the matrix \( S \):

\[
S = \begin{bmatrix}
x & y^2 & z^3 \\
1 & 0 & 1 \\
3 + x & 2 - y^2 & 4z^3
\end{bmatrix}
\]

You are also given another matrix:

\[
\begin{bmatrix}
-x & -z^3 & -y^2 \\
1 & 1 & 0 \\
3 + x & 4z^3 & 2 - y^2
\end{bmatrix}
\]

The task seems to focus on evaluating the properties of \( S \) to help solve for unknowns given that \( |S| = 6 \). 

Let me proceed to compute the determinant of \( S \) symbolically using its elements.The determinant of the matrix \( S \) is:

\[
|S| = 2xy^2 - 2x - 5y^2z^3 + 3y^2 + 2z^3
\]

Given that \( |S| = 6 \), we can set the expression equal to 6:

\[
2xy^2 - 2x - 5y^2z^3 + 3y^2 + 2z^3 = 6
\]

From here, you can solve this equation for any given values of \( x \), \( y \), and \( z \), or if needed, manipulate it further to isolate a specific variable.

Would you like to proceed with a particular value for any of these variables, or should I attempt to simplify it further?

---

**Follow-up questions:**

1. Would you like to solve for a specific variable, say \( x \), \( y \), or \( z \)?
2. Do you need to understand how determinants are computed for 3x3 matrices?
3. Should we try plugging in specific values to explore potential solutions?
4. Are there any constraints or specific cases you would like to explore for the variables?
5. Would you like to verify the properties of the second matrix mentioned?

**Tip:** The determinant of a matrix can provide key insights into whether a system of linear equations has a unique solution (when non-zero) or infinitely many/no solutions (when zero).

Given that matrix \( S \) and \( |S| = 6 \), find the determinant of the matrix \( S \).

Learn how to calculate the determinant of a 3x3 matrix given matrix \( S \) and \( |S| = 6 \). This example covers core linear algebra concepts such as matrix determinant formulas and key theorems. Suitable for students at the undergraduate level.

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