If `|(3x-8,3,3),(3,3x-8,3),(3,3,3x-8)|` = 0 then x =

Options

A. `8//3`
B. `2//3`
C. `1//3`
D. none

Answer

To solve the determinant \( D = \begin{vmatrix} 3x - 8 & 3 & 3 \\ 3 & 3x - 8 & 3 \\ 3 & 3 & 3x - 8 \end{vmatrix} = 0 \), we will follow these steps:

### Step 1: Apply Column Operation
We will perform the column operation \( C_1 \rightarrow C_1 + C_2 + C_3 \). This means we will add the elements of columns \( C_2 \) and \( C_3 \) to the elements of column \( C_1 \).

After applying this operation, we get:
\[ D = \begin{vmatrix} (3x - 8) + 3 + 3 & 3 & 3 \\ 3 + (3x - 8) + 3 & (3x - 8) & 3 \\ 3 + 3 + (3x - 8) & 3 & (3x - 8) \end{vmatrix} \]
This simplifies to:
\[ D = \begin{vmatrix} 3x - 2 & 3 & 3 \\ 3x - 2 & 3x - 8 & 3 \\ 3x - 2 & 3 & 3x - 8 \end{vmatrix} \]

### Step 2: Factor Out Common Terms
Notice that the first column has a common factor of \( 3x - 2 \). We can factor this out:
\[ D = (3x - 2) \begin{vmatrix} 1 & 3 & 3 \\ 1 & 3x - 8 & 3 \\ 1 & 3 & 3x - 8 \end{vmatrix} \]

### Step 3: Apply Row Operations
Next, we will apply row operations to simplify the determinant. We can perform \( R_2 \rightarrow R_2 - R_1 \) and \( R_3 \rightarrow R_3 - R_1 \):
\[ D = (3x - 2) \begin{vmatrix} 1 & 3 & 3 \\ 0 & (3x - 8) - 3 & 0 \\ 0 & 0 & (3x - 8) - 3 \end{vmatrix} \]
This simplifies to:
\[ D = (3x - 2) \begin{vmatrix} 1 & 3 & 3 \\ 0 & 3x - 11 & 0 \\ 0 & 0 & 3x - 11 \end{vmatrix} \]

### Step 4: Calculate the Determinant
The determinant of a matrix with two rows of zeros is simply the product of the diagonal elements:
\[ D = (3x - 2) \cdot (3x - 11) \cdot 1 \]

### Step 5: Set the Determinant to Zero
Now we set the determinant equal to zero:
\[ (3x - 2)(3x - 11) = 0 \]
This gives us two equations to solve:
1. \( 3x - 2 = 0 \)
2. \( 3x - 11 = 0 \)

### Step 6: Solve for \( x \)
From the first equation:
\[ 3x = 2 \implies x = \frac{2}{3} \]
From the second equation:
\[ 3x = 11 \implies x = \frac{11}{3} \]

### Final Answer
Thus, the values of \( x \) are:
\[ x = \frac{2}{3} \quad \text{or} \quad x = \frac{11}{3} \]

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