https://file.aiquickdraw.com/mathbot/file/173316131852624xy4sk7.jpg

Yes, let’s carefully check if there are any mistakes in the determinant calculation provided in the image.

The determinant of the \(3 \times 3\) matrix \( A = \begin{pmatrix} 1 & 2 & -1 \\ 1 & 2 & 1 \\ 0 & 1 & -2 \end{pmatrix} \) is computed using cofactor expansion along the first row:

\[
|A| = 1 \cdot \begin{vmatrix} 2 & 1 \\ 1 & -2 \end{vmatrix} - 2 \cdot \begin{vmatrix} 1 & 1 \\ 0 & -2 \end{vmatrix} + (-1) \cdot \begin{vmatrix} 1 & 2 \\ 0 & 1 \end{vmatrix}
\]

Step-by-step:

1. **First Minor**:
   \[
   \begin{vmatrix} 2 & 1 \\ 1 & -2 \end{vmatrix} = (2)(-2) - (1)(1) = -4 - 1 = -5
   \]

2. **Second Minor**:
   \[
   \begin{vmatrix} 1 & 1 \\ 0 & -2 \end{vmatrix} = (1)(-2) - (1)(0) = -2
   \]

3. **Third Minor**:
   \[
   \begin{vmatrix} 1 & 2 \\ 0 & 1 \end{vmatrix} = (1)(1) - (2)(0) = 1
   \]

Substitute into the determinant formula:

\[
|A| = 1(-5) - 2(-2) + (-1)(1)
\]

Simplify:

\[
|A| = -5 + 4 - 1 = -2
\]

The calculation in the image matches this result, so **there is no mistake** in the determinant calculation. 

If you have further questions or want a clarification, feel free to ask!

### Follow-up Questions:
1. What does the determinant of a matrix represent geometrically?
2. How do you compute determinants for matrices larger than \(3 \times 3\)?
3. Why is cofactor expansion valid for determinant computation?
4. Can the determinant be zero for a non-zero matrix? Why?
5. How does the determinant relate to matrix invertibility?

### Tip:
Always double-check cofactor signs (+/-) when expanding along a row or column to avoid errors.

Verify the determinant calculation of a 3x3 matrix using cofactor expansion. Explore linear algebra concepts, step-by-step computation, and determinant properties suitable for advanced high school or undergraduate students.

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