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:

   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:

   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:

   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.