Let `A=[{:(-5,-8,-7),(3,5,4),(2,3,3):}]` and `B=[{:(x),(y),(2):}]`. If `AB` is a scalar multiple of `B`, then the value of `x+y` is

Options

A. `-1`
B. `-2`
C. `1`
D. `2`

Explanation

To solve the problem, we need to find the value of \( x + y \) given that the product of matrices \( A \) and \( B \) is a scalar multiple of \( B \).

### Step-by-Step Solution:

1. **Define the matrices**:
\[ A = \begin{pmatrix} -5 & -8 & -7 \\ 3 & 5 & 4 \\ 2 & 3 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} x \\ y \\ 2 \end{pmatrix} \]

2. **Calculate the product \( AB \)**:
\[ AB = A \cdot B = \begin{pmatrix} -5 & -8 & -7 \\ 3 & 5 & 4 \\ 2 & 3 & 3 \end{pmatrix} \begin{pmatrix} x \\ y \\ 2 \end{pmatrix} \]

Performing the matrix multiplication:
- First row:
\[ -5x - 8y - 14 \]
- Second row:
\[ 3x + 5y + 8 \]
- Third row:
\[ 2x + 3y + 6 \]

Thus, we have:
\[ AB = \begin{pmatrix} -5x - 8y - 14 \\ 3x + 5y + 8 \\ 2x + 3y + 6 \end{pmatrix} \]

3. **Set up the equation**:
Since \( AB \) is a scalar multiple of \( B \), we can write:
\[ AB = \lambda B = \lambda \begin{pmatrix} x \\ y \\ 2 \end{pmatrix} \]
This gives us the following equations:
- From the first row:
\[ -5x - 8y - 14 = \lambda x \quad (1) \]
- From the second row:
\[ 3x + 5y + 8 = \lambda y \quad (2) \]
- From the third row:
\[ 2x + 3y + 6 = 2\lambda \quad (3) \]

4. **Rearranging the equations**:
Rearranging each equation gives:
- Equation (1):
\[ \lambda x + 5x + 8y + 14 = 0 \quad (4) \]
- Equation (2):
\[ \lambda y - 3x - 5y - 8 = 0 \quad (5) \]
- Equation (3):
\[ 2\lambda - 2x - 3y - 6 = 0 \quad (6) \]

5. **Add equations (4), (5), and (6)**:
Adding all three equations:
\[ (\lambda x + 5x + 8y + 14) + (\lambda y - 3x - 5y - 8) + (2\lambda - 2x - 3y - 6) = 0 \]
This simplifies to:
\[ 0 = \lambda(x + y + 2) \]

6. **Conclusion**:
Since \( \lambda \) is a non-zero scalar, we have:
\[ x + y + 2 = 0 \implies x + y = -2 \]

### Final Answer:
The value of \( x + y \) is \( -2 \).