https://file.aiquickdraw.com/mathbot/file/1729886315580t7nr44gv.jpg

The problem asks for the determinant of matrix \( A \) after performing Gaussian elimination. Based on the image, the elimination involves a sequence of elementary row operations.

Let's break down the key steps:

1. The original matrix \( A \) is:
   \[
   A = \begin{bmatrix}
   1 & 0 & 3 \\
   0 & 1 & 0 \\
   0 & 0 & 1
   \end{bmatrix}
   \]

2. **First operation \( E_1 \):** No row swapping or scaling seems to have been done in this case.

3. **Second operation \( E_2 \):** The matrix changes to:
   \[
   E_2 A = \begin{bmatrix}
   1 & 0 & 0 \\
   0 & 7 & 0 \\
   0 & 0 & 1
   \end{bmatrix}
   \]
   This looks like a multiplication of the second row by 7. 

4. **Third operation \( E_3 \):** The final matrix becomes:
   \[
   E_3 E_2 A = \begin{bmatrix}
   1 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 1
   \end{bmatrix}
   \]
   The second row is divided by 7, effectively restoring the identity matrix.

### Determinant Calculation:

- The determinant of the original matrix \( A \) is affected by these operations:
   - Scaling the second row by 7 multiplies the determinant by 7.
   - Dividing the second row by 7 divides the determinant by 7.

Hence, the determinant of \( A \) remains unchanged after these operations. Since \( A \) is initially a triangular matrix with diagonal elements 1, 1, and 1, its determinant is:

\[
\text{det}(A) = 1
\]

Would you like any further details or explanations on the process?

### Follow-up Questions:
1. What is the determinant of a matrix when an elementary row swap is involved?
2. How does multiplying a row by a scalar affect the determinant?
3. What is the determinant of a diagonal matrix?
4. How can the determinant be used to check for matrix invertibility?
5. What are the steps for Gaussian elimination?

### Tip:
The determinant of an upper or lower triangular matrix is simply the product of its diagonal elements.

Consider the following Gauss elimination and determine the determinant of matrix A after the operations.

Learn how to find the determinant of a matrix using Gaussian elimination. This problem explores elementary row operations and determinant properties, suitable for undergraduate linear algebra students.

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