If `|{:(4,1),(2,1):}|^(2)=|{:(3,2),(1,x):}|-|{:(x,3),(-2,1):}|`, then the value of x is :

Options

A. 5
B. 6
C. 8
D. 9

Answer

To solve the equation given in the problem, we need to evaluate the determinants of the matrices and set up the equation accordingly.

### Step-by-Step Solution:

1. **Identify the Matrices:**
We have three matrices:
- Matrix A: \(\begin{pmatrix} 4 & 1 \\ 2 & 1 \end{pmatrix}\)
- Matrix B: \(\begin{pmatrix} 3 & 2 \\ 1 & x \end{pmatrix}\)
- Matrix C: \(\begin{pmatrix} x & 3 \\ -2 & 1 \end{pmatrix}\)

2. **Calculate the Determinant of Matrix A:**
The determinant of a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is given by \(ad - bc\).
\[ \text{Det}(A) = 4 \cdot 1 - 1 \cdot 2 = 4 - 2 = 2 \]

3. **Calculate the Determinant of Matrix B:**
\[ \text{Det}(B) = 3 \cdot x - 2 \cdot 1 = 3x - 2 \]

4. **Calculate the Determinant of Matrix C:**
\[ \text{Det}(C) = x \cdot 1 - 3 \cdot (-2) = x + 6 \]

5. **Set Up the Equation:**
According to the problem, we have:
\[ |A|^2 = |B| - |C| \]
Substituting the determinants we calculated:
\[ 2^2 = (3x - 2) - (x + 6) \]
Simplifying this gives:
\[ 4 = 3x - 2 - x - 6 \]

6. **Combine Like Terms:**
\[ 4 = 2x - 8 \]

7. **Isolate x:**
Add 8 to both sides:
\[ 4 + 8 = 2x \implies 12 = 2x \]
Divide by 2:
\[ x = 6 \]

### Final Answer:
The value of \(x\) is \(6\).