[1 x 1 ] `[{:(1, 3,2),(2, 5, 1),(15, 3, 2):}] [{:(1),(2),(x):}]` = 0, if x =

Options

A. `- 7 `
B. `-11`
C. `-2`
D. 14

Answer

To solve the equation given by the matrix multiplication, we will follow these steps:

### Step 1: Understand the matrices involved
We have a 3x3 matrix \( A \) and a 3x1 matrix \( B \):
\[ A = \begin{pmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 2 \\ x \end{pmatrix} \]

### Step 2: Perform the matrix multiplication
We need to multiply matrix \( A \) by matrix \( B \):
\[ AB = \begin{pmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \\ x \end{pmatrix} \]

The result will be a 3x1 matrix. We compute each entry in the resulting matrix:

- First entry:
\[ 1 \cdot 1 + 3 \cdot 2 + 2 \cdot x = 1 + 6 + 2x = 7 + 2x \]

- Second entry:
\[ 2 \cdot 1 + 5 \cdot 2 + 1 \cdot x = 2 + 10 + x = 12 + x \]

- Third entry:
\[ 15 \cdot 1 + 3 \cdot 2 + 2 \cdot x = 15 + 6 + 2x = 21 + 2x \]

Thus, we have:
\[ AB = \begin{pmatrix} 7 + 2x \\ 12 + x \\ 21 + 2x \end{pmatrix} \]

### Step 3: Set the resulting matrix equal to zero
We want the resulting matrix to equal the zero matrix:
\[ \begin{pmatrix} 7 + 2x \\ 12 + x \\ 21 + 2x \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \]

This gives us three equations to solve:
1. \( 7 + 2x = 0 \)
2. \( 12 + x = 0 \)
3. \( 21 + 2x = 0 \)

### Step 4: Solve each equation
1. From \( 7 + 2x = 0 \):
\[ 2x = -7 \implies x = -\frac{7}{2} \]

2. From \( 12 + x = 0 \):
\[ x = -12 \]

3. From \( 21 + 2x = 0 \):
\[ 2x = -21 \implies x = -\frac{21}{2} \]

### Step 5: Collect the solutions
The values of \( x \) that satisfy the equations are:
- \( x = -\frac{7}{2} \)
- \( x = -12 \)
- \( x = -\frac{21}{2} \)

### Final Result
The values of \( x \) for which the matrix multiplication equals zero are:
\[ x = -\frac{7}{2}, \quad x = -12, \quad x = -\frac{21}{2} \]