If `A({:(1,3,4),(3,-1,5),(-2,4,-3):})=({:(3,-1,5),(1,3,4),(+4,-8,6):})`, then `A=`

Options

A. `({:(1,0,0),(0,1,0),(0,0,-2):})`
B. `({:(0,1,0),(1,0,0),(0,0,1):})`
C. `({:(1,0,0),(1,0,0),(0,0,-2):})`
D. `({:(0,1,0),(1,0,0),(0,0,-2):})`

Explanation

To solve the problem, we need to find the matrix \( A \) such that:

\[ A \cdot \begin{pmatrix} 1 & 3 & 4 \\ 3 & -1 & 5 \\ -2 & 4 & -3 \end{pmatrix} = \begin{pmatrix} 3 & -1 & 5 \\ 1 & 3 & 4 \\ 4 & -8 & 6 \end{pmatrix} \]

### Step 1: Identify the transformation between the matrices

We notice that the rows of the resulting matrix on the right side can be obtained from the rows of the matrix on the left side through some operations. Specifically, we can observe that:

- The first row of the result (3, -1, 5) corresponds to the second row of the original matrix (3, -1, 5).
- The second row of the result (1, 3, 4) corresponds to the first row of the original matrix (1, 3, 4).
- The third row of the result (4, -8, 6) can be obtained by multiplying the third row of the original matrix (-2, 4, -3) by -2.

### Step 2: Determine the row operations

From the observations, we can summarize the operations as follows:
- Interchange the first and second rows.
- Multiply the third row by -2.

### Step 3: Construct the transformation matrix \( A \)

To perform these operations using a transformation matrix \( A \), we need to create a matrix that reflects these row operations. The identity matrix \( I \) is:

\[ I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

Now, we will modify \( I \) to reflect our row operations:
1. Interchange rows 1 and 2.
2. Multiply row 3 by -2.

This gives us:

\[ A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -2 \end{pmatrix} \]

### Step 4: Write the final answer

Thus, the matrix \( A \) is:

\[ A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -2 \end{pmatrix} \]