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To solve this problem, let's proceed with the cofactor expansion across the first row, as requested.

The matrix is:
\[
\begin{bmatrix}
3 & -2 & 3 \\
2 & 1 & 2 \\
1 & 5 & -2 \\
\end{bmatrix}
\]

### Step 1: Determinant Using Cofactor Expansion Across the First Row

The determinant of a \(3 \times 3\) matrix \(A\) can be calculated using cofactor expansion along the first row:
\[
\text{det}(A) = a_{11} \cdot C_{11} + a_{12} \cdot C_{12} + a_{13} \cdot C_{13}
\]
where \(C_{ij}\) is the cofactor of the element in the \(i\)-th row and \(j\)-th column.

For this matrix:
1. \(a_{11} = 3\), and the minor matrix for \(a_{11}\) is:
   \[
   \begin{vmatrix}
   1 & 2 \\
   5 & -2 \\
   \end{vmatrix} = (1)(-2) - (2)(5) = -2 - 10 = -12
   \]
   Therefore, \(C_{11} = (-1)^{1+1} \cdot (-12) = -12\).

2. \(a_{12} = -2\), and the minor matrix for \(a_{12}\) is:
   \[
   \begin{vmatrix}
   2 & 2 \\
   1 & -2 \\
   \end{vmatrix} = (2)(-2) - (2)(1) = -4 - 2 = -6
   \]
   Therefore, \(C_{12} = (-1)^{1+2} \cdot (-6) = 6\).

3. \(a_{13} = 3\), and the minor matrix for \(a_{13}\) is:
   \[
   \begin{vmatrix}
   2 & 1 \\
   1 & 5 \\
   \end{vmatrix} = (2)(5) - (1)(1) = 10 - 1 = 9
   \]
   Therefore, \(C_{13} = (-1)^{1+3} \cdot 9 = 9\).

Now we can calculate the determinant:
\[
\text{det}(A) = 3(-12) + (-2)(6) + 3(9) = -36 - 12 + 27 = -21
\]

### Step 2: Determine the Correct Answer from Choices

From the answer choices given:
- Option A: \(-3(-12) + (-2)(-6) - (3)(9)\)
- Option B: \((-2)(-6) + (1)(-9) - (5)(0)\)
- Option C: \(3(-12) - (-2)(-6) + (3)(9)\)
- Option D: \((-2)(-6) - (1)(-9) + (5)(0)\)

Our expansion matches **Option C**:
\[
3(-12) - (-2)(-6) + (3)(9) = -21
\]

So, the correct answer is **Option C** with a result of **-21**.

Would you like further details, or do you have any questions?

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Here are some related questions for further exploration:
1. What is the purpose of cofactor expansion in finding determinants?
2. How do we select which row or column to expand for easier calculations?
3. What are other methods to compute the determinant of a \(3 \times 3\) matrix?
4. How does the cofactor sign pattern affect determinant calculation?
5. How does the determinant value impact matrix invertibility?

**Tip:** When choosing rows or columns for cofactor expansion, select those with the most zeroes for simplified calculations.

Compute the determinant using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.
Select the correct answer choice from the provided options.

Learn how to compute the determinant of a 3x3 matrix using cofactor expansion across the first row. This example includes step-by-step calculations of minors, cofactors, and the final determinant. Suitable for high school students learning matrix algebra.

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