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 equation \( A \cdot \begin{pmatrix} 1 & 3 & 4 \\ 3 & -1 & 5 \\ -2 & 4 & -3 \end{pmatrix} = \begin{pmatrix} 3 & 1 & 4 \\ -1 & 3 & -8 \\ 5 & 4 & 6 \end{pmatrix} \), we need to find the matrix \( A \).

### Step-by-Step Solution:

1. **Write down the equation**:
\[ A \cdot B = C \]
where
\[ B = \begin{pmatrix} 1 & 3 & 4 \\ 3 & -1 & 5 \\ -2 & 4 & -3 \end{pmatrix} \]
and
\[ C = \begin{pmatrix} 3 & 1 & 4 \\ -1 & 3 & -8 \\ 5 & 4 & 6 \end{pmatrix} \]

2. **Identify the dimensions**:
Matrix \( A \) must be a \( 3 \times 3 \) matrix since \( B \) is \( 3 \times 3 \) and \( C \) is \( 3 \times 3 \).

3. **Assume a general form for \( A \)**:
Let
\[ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

4. **Multiply \( A \) and \( B \)**:
The product \( A \cdot B \) is calculated as follows:
\[ A \cdot B = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \cdot \begin{pmatrix} 1 & 3 & 4 \\ 3 & -1 & 5 \\ -2 & 4 & -3 \end{pmatrix} \]
This results in:
\[ = \begin{pmatrix} a(1) + b(3) + c(-2) & a(3) + b(-1) + c(4) & a(4) + b(5) + c(-3) \\ d(1) + e(3) + f(-2) & d(3) + e(-1) + f(4) & d(4) + e(5) + f(-3) \\ g(1) + h(3) + i(-2) & g(3) + h(-1) + i(4) & g(4) + h(5) + i(-3) \end{pmatrix} \]

5. **Set the equations equal to the corresponding elements of matrix \( C \)**:
From the first row:
\[ a + 3b - 2c = 3 \quad (1) \]
\[ 3a - b + 4c = 1 \quad (2) \]
\[ 4a + 5b - 3c = 4 \quad (3) \]
From the second row:
\[ d + 3e - 2f = -1 \quad (4) \]
\[ 3d - e + 4f = 3 \quad (5) \]
\[ 4d + 5e - 3f = -8 \quad (6) \]
From the third row:
\[ g + 3h - 2i = 5 \quad (7) \]
\[ 3g - h + 4i = 4 \quad (8) \]
\[ 4g + 5h - 3i = 6 \quad (9) \]

6. **Solve the system of equations**:
We will solve equations (1), (2), and (3) for \( a, b, c \) and equations (4), (5), and (6) for \( d, e, f \) and equations (7), (8), and (9) for \( g, h, i \).

After solving these equations, we find:
\[ A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & -2 & -2 \end{pmatrix} \]

### Final Result:
Thus, the matrix \( A \) is:
\[ A = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & -2 & -2 \end{pmatrix} \]