If `[(1,x)][(2,-1),(1,2)][(1),(3)]=[0]`, then x=

Options

A. `2/7`
B. `1/7`
C. `5/7`
D. `3/7`

Answer

To solve the equation given in the question, we need to perform matrix multiplication and set the resulting matrix equal to the zero matrix. Let's break it down step by step.

### Step-by-Step Solution:

1. **Identify the Matrices:**
The given matrices are:
- Row matrix: \( R = [1, x] \)
- 2x2 matrix: \( A = \begin{bmatrix} 2 & -1 \\ 1 & 2 \end{bmatrix} \)
- Column matrix: \( C = \begin{bmatrix} 1 \\ 3 \end{bmatrix} \)

2. **Matrix Multiplication:**
We need to multiply the row matrix \( R \) with the 2x2 matrix \( A \) first, and then multiply the result with the column matrix \( C \).

First, compute \( R \cdot A \):
\[ R \cdot A = [1, x] \cdot \begin{bmatrix} 2 & -1 \\ 1 & 2 \end{bmatrix} \]
The result will be a row matrix with two elements:
- First element: \( 1 \cdot 2 + x \cdot 1 = 2 + x \)
- Second element: \( 1 \cdot (-1) + x \cdot 2 = -1 + 2x \)

Therefore,
\[ R \cdot A = [2 + x, -1 + 2x] \]

3. **Multiply the Result with the Column Matrix:**
Now, we multiply the resulting row matrix with the column matrix \( C \):
\[ (R \cdot A) \cdot C = [2 + x, -1 + 2x] \cdot \begin{bmatrix} 1 \\ 3 \end{bmatrix} \]
The result will be a single element:
\[ (2 + x) \cdot 1 + (-1 + 2x) \cdot 3 = 2 + x - 3 + 6x = 2 + x - 3 + 6x = 7x - 1 \]

4. **Set the Result Equal to Zero:**
According to the problem, this result equals the zero matrix:
\[ 7x - 1 = 0 \]

5. **Solve for \( x \):**
Rearranging the equation gives:
\[ 7x = 1 \]
Dividing both sides by 7, we find:
\[ x = \frac{1}{7} \]

### Final Answer:
Thus, the value of \( x \) is \( \frac{1}{7} \).

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