Let's go step by step to determine the answers.

Step 1: Finding the first row operation

We are given matrix $A$:
$A = \begin{bmatrix} 2 & -1 & 0 \\ -6 & 4 & -1 \\ 4 & -6 & 7 \end{bmatrix}$ And after the first row operation, we get matrix $B$:
$B = \begin{bmatrix} 2 & -1 & 0 \\ 0 & 1 & -1 \\ 4 & -6 & 7 \end{bmatrix}$

The second row ($R_2$) was replaced by: $R_2 \to R_2 + k R_1$ Looking at the first column, to eliminate the $-6$ in $R_2$, we need: $-6 + k(2) = 0$ Solving for $k$: $k = \frac{6}{2} = 3$

Thus, the row operation is: $R_2 \to R_2 + 3 R_1$

Answer 1 Question 2: 3

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Step 2: Finding the second row operation

Now, we move from $B$ to $C$:
$C = \begin{bmatrix} 2 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 3 \end{bmatrix}$

The third row ($R_3$) was replaced by: $R_3 \to R_3 + m R_2$ Looking at the second column, to eliminate the $-6$ in $R_3$, we need: $-6 + m(1) = 0$ Solving for $m$: $m = 6$

Thus, the row operation is: $R_3 \to R_3 + 6 R_2$

Answer 3 Question 2: 6

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Step 3: Elementary Matrix $E_1$

The elementary matrix $E_1$ corresponds to the operation $R_2 \to R_2 + 3 R_1$, so:

$E_1 = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

• (2,1)-entry of $E_1$: 3
• (3,1)-entry of $E_1$: 0
• (3,2)-entry of $E_1$: 0

Answers:
Answer 4 Question 2: 3
Answer 5 Question 2: 0
Answer 6 Question 2: 0

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Step 4: Elementary Matrix $E_2$

The elementary matrix $E_2$ corresponds to $R_3 \to R_3 + 6 R_2$, so:

$E_2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 6 & 1 \end{bmatrix}$

• (2,1)-entry of $E_2$: 0
• (3,1)-entry of $E_2$: 0
• (3,2)-entry of $E_2$: 6

Answers:
Answer 7 Question 2: 0
Answer 8 Question 2: 0
Answer 9 Question 2: 6

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Step 5: Elementary Matrix $E_3$

The problem statement mentions $D$, but $D$ is not defined in the question. Since we have already obtained an upper triangular form in $C$, no further row operations are needed. Thus, $E_3$ is just the identity matrix:

$E_3 = I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

• (2,1)-entry of $E_3$: 0
• (3,1)-entry of $E_3$: 0
• (3,2)-entry of $E_3$: 0

Answers:
Answer 10 Question 2: 0
Answer 11 Question 2: 0
Answer 12 Question 2: 0

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Summary of Answers:

1. Answer 1 Question 2: 3
2. Answer 3 Question 2: 6
3. Answer 4 Question 2: 3
4. Answer 5 Question 2: 0
5. Answer 6 Question 2: 0
6. Answer 7 Question 2: 0
7. Answer 8 Question 2: 0
8. Answer 9 Question 2: 6
9. Answer 10 Question 2: 0
10. Answer 11 Question 2: 0
11. Answer 12 Question 2: 0

Would you like a more detailed explanation or verification of any step?

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Further Exploration:

1. How do we determine elementary matrices for row operations?
2. How can we use elementary matrices to compute the determinant of $A$?
3. What is the significance of upper triangular matrices in solving linear equations?
4. How do row operations affect the rank of a matrix?
5. What happens if a pivot in the elimination process is zero?

Tip:

When performing row operations, always ensure that elementary matrices correspond exactly to the transformations applied, as they represent the identity matrix with a single modified row.