Let's go through each row operation shown in the matrix, step-by-step, to complete the missing entries.

The matrix initially given is:

$\begin{bmatrix} 4 & -3 & 6 \\ -4 & 5 & -4 \end{bmatrix}$

Step 1: Applying $-4R_1$

The operation $-4R_1$ means we multiply each element in the first row $R_1$ by $-4$:

1. For the element in the first row and first column: $4 \times -4 = -16$
2. For the element in the first row and second column: $-3 \times -4 = 12$
3. For the element in the first row and third column: $6 \times -4 = -24$

After performing this operation, the new matrix becomes:

$\begin{bmatrix} -16 & 12 & -24 \\ -4 & 5 & -4 \end{bmatrix}$

Step 2: Applying $R_2 + 3R_1$

For this step, we add $3$ times the first row $R_1$ to the second row $R_2$:

1. For the element in the second row and first column: $-4 + 3 \times 4 = -4 + 12 = 8$
2. For the element in the second row and second column: $5 + 3 \times -3 = 5 - 9 = -4$
3. For the element in the second row and third column: $-4 + 3 \times 6 = -4 + 18 = 14$

After this operation, the matrix becomes:

$\begin{bmatrix} 4 & -3 & 6 \\ 8 & -4 & 14 \end{bmatrix}$

Step 3: Applying $R_1 - 4R_2$

In this operation, we subtract $4$ times the second row $R_2$ from the first row $R_1$:

1. For the element in the first row and first column: $4 - 4 \times 8 = 4 - 32 = -28$
2. For the element in the first row and second column: $-3 - 4 \times -4 = -3 + 16 = 13$
3. For the element in the first row and third column: $6 - 4 \times 14 = 6 - 56 = -50$

After this operation, the matrix becomes:

$\begin{bmatrix} -28 & 13 & -50 \\ 8 & -4 & 14 \end{bmatrix}$

Summary of the Results

The resulting matrices after each row operation are:

1. After $-4R_1$: $\begin{bmatrix} -16 & 12 & -24 \\ -4 & 5 & -4 \end{bmatrix}$

2. After $R_2 + 3R_1$: $\begin{bmatrix} 4 & -3 & 6 \\ 8 & -4 & 14 \end{bmatrix}$

3. After $R_1 - 4R_2$: $\begin{bmatrix} -28 & 13 & -50 \\ 8 & -4 & 14 \end{bmatrix}$

Would you like further explanation on any of the steps?

Relative Questions:

1. What is the purpose of row operations in matrix transformations?
2. How does multiplying a row by a scalar affect the determinant of the matrix?
3. What is the significance of the identity matrix in row operations?
4. Can matrix row operations change the rank of a matrix? Why or why not?
5. How do row operations help in solving systems of linear equations?

Tip:

Row operations are fundamental for simplifying matrices in Gaussian elimination, which helps solve systems of linear equations efficiently.