Given the matrix � = [ − 2 − 4 8 6 10 7 4 12 5 ] A= ​ −2 6 4 ​ −4 10 12 ​ 8 7 5 ​ ​ , evaluate the determinant ∣ � ∣ ∣A∣. Use elementary row operations to evaluate |A|

To find the determinant of the matrix \( A = \begin{bmatrix} -2 & 6 & 4 \\ -4 & 10 & 12 \\ 8 & 7 & 5 \end{bmatrix} \), we will use elementary row operations to simplify the matrix and calculate its determinant.

### Step 1: Start with the matrix \( A \)
\[
A = \begin{bmatrix} -2 & 6 & 4 \\ -4 & 10 & 12 \\ 8 & 7 & 5 \end{bmatrix}
\]

### Step 2: Use elementary row operations to simplify the matrix
We will aim to transform the matrix into an upper triangular form, where all elements below the main diagonal are zero. The determinant of a triangular matrix is simply the product of the diagonal entries.

#### Row Operation 1: Make the element \( a_{21} \) (second row, first column) zero
To eliminate \( -4 \) in the second row, we can replace row 2 with:
\[
R_2 \to R_2 - 2R_1
\]
This gives:
\[
\begin{bmatrix}
-2 & 6 & 4 \\
0 & -2 & 4 \\
8 & 7 & 5
\end{bmatrix}
\]

#### Row Operation 2: Make the element \( a_{31} \) (third row, first column) zero
To eliminate \( 8 \) in the third row, we can replace row 3 with:
\[
R_3 \to R_3 + 4R_1
\]
This gives:
\[
\begin{bmatrix}
-2 & 6 & 4 \\
0 & -2 & 4 \\
0 & 31 & 21
\end{bmatrix}
\]

#### Row Operation 3: Make the element \( a_{32} \) (third row, second column) zero
To eliminate \( 31 \) in the third row, we can replace row 3 with:
\[
R_3 \to R_3 + \frac{31}{2} R_2
\]
This gives:
\[
\begin{bmatrix}
-2 & 6 & 4 \\
0 & -2 & 4 \\
0 & 0 & 83
\end{bmatrix}
\]

### Step 3: Compute the determinant
Now that the matrix is in upper triangular form, we can compute the determinant as the product of the diagonal elements:
\[
\det(A) = (-2) \times (-2) \times 83 = 4 \times 83 = 332
\]

### Final Answer:
\[
\boxed{\det(A) = 332}
\]

Would you like more details on any step? 

Here are 5 follow-up questions to expand on this topic:
1. How does the determinant change if we multiply a row by a scalar?
2. What are the properties of determinants with regard to matrix operations?
3. How can elementary row operations affect the determinant?
4. How can you use cofactor expansion to find the determinant of a matrix?
5. How can you interpret the geometric meaning of the determinant?

**Tip**: When performing row operations, swapping two rows changes the sign of the determinant!

Given the matrix A = [ −2 −4 8 6 10 7 4 12 5 ], evaluate the determinant |A|. Use elementary row operations to evaluate |A|.

Learn how to evaluate the determinant of a 3x3 matrix using elementary row operations. This step-by-step guide shows how to transform the matrix into an upper triangular form and calculate the determinant. Suitable for advanced high school or undergraduate students.

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