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Solve this problem by elementary transformation 

To solve the problem of finding the value of \( x \) using elementary row transformations, we'll work with the given matrix:

\[
\begin{pmatrix}
1 & 1 & -2 & x \\
2 & x^2 & -x & 3x-3 \\
3 & 2x+2 & -3 & 2x+3
\end{pmatrix}
\]

We aim to use row operations to transform this matrix to an upper triangular form, where the determinant is the product of the diagonal elements. Then, we can set the determinant equal to zero and solve for \( x \).

### Steps for Elementary Row Operations:
1. **Make the first column's first element 1:** It’s already 1, so no changes needed.
  
2. **Eliminate the elements below the first pivot (first column, second and third rows):**
   - Row 2: Subtract 2 times Row 1 from Row 2.
   - Row 3: Subtract 3 times Row 1 from Row 3.

3. **Eliminate the elements below the second pivot (second column, third row):**
   - Row 3: Adjust Row 3 based on the modified Row 2.

Let's perform these row operations.

### Row Operations

1. **Row 2 transformation:**
   \[
   R_2 \rightarrow R_2 - 2R_1
   \]
2. **Row 3 transformation:**
   \[
   R_3 \rightarrow R_3 - 3R_1
   \]
3. **Row 3 transformation based on Row 2:**
   \[
   R_3 \rightarrow R_3 - \left(\frac{\text{coefficient}}{\text{pivot}}\right)R_2
   \]

After these steps, we’ll compute the determinant and set it to zero to find \( x \).

I'll perform these calculations next.After performing the elementary row transformations, the matrix is transformed into:

\[
\begin{pmatrix}
1 & 1 & -2 & x \\
0 & x^2 - 2 & 4 - x & x - 3 \\
0 & 0 & \frac{-(4 - x)(2x - 1)}{x^2 - 2} + 3 & -x - \frac{(x - 3)(2x - 1)}{x^2 - 2} + 3
\end{pmatrix}
\]

The determinant of this matrix, which is the product of the diagonal elements in the upper triangular form, simplifies to:

\[
\text{Determinant} = (x^2 - 2) \times \left(\frac{-(4 - x)(2x - 1)}{x^2 - 2} + 3\right)
\]

Setting this determinant equal to zero gives the solutions:

\[
x = -\frac{1}{5} \quad \text{or} \quad x = 2
\]

These are the values of \( x \) that make the determinant of the matrix zero, consistent with our previous method.

Would you like any further details or have any questions?

### Related Questions:
1. What are the conditions for a matrix to be singular (non-invertible)?
2. How can elementary row operations help in solving systems of linear equations?
3. What is the geometric interpretation of elementary row transformations?
4. How does the determinant change under row operations?
5. How can the determinant be used to check the consistency of a system of linear equations?

### Tip:
When performing elementary row transformations, it's crucial to keep track of how each step affects the determinant. Specifically, swapping rows changes the sign of the determinant, while scaling a row scales the determinant by the same factor.

Learn how to solve a matrix problem using elementary row transformations to calculate the determinant. This detailed solution involves applying row operations to transform the matrix to upper triangular form and using the determinant to find the value of x. Suitable for advanced high school students and above.

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