The roots of the equation `|{:(x-1,1,1),(1,x-1,1),(1,1,x-1):}|=0` are

Options

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

Answer

To solve the equation given by the determinant \(|{:(x-1,1,1),(1,x-1,1),(1,1,x-1):}|=0\), we will follow these steps:

### Step 1: Write the determinant in a simplified form
We start with the determinant:
\[ D = \begin{vmatrix} x-1 & 1 & 1 \\ 1 & x-1 & 1 \\ 1 & 1 & x-1 \end{vmatrix} \]

### Step 2: Perform row operations
We can simplify the determinant by performing row operations. Let's subtract the third row from the second row:
\[ R_2 \rightarrow R_2 - R_3 \]
This gives us:
\[ D = \begin{vmatrix} x-1 & 1 & 1 \\ 0 & (x-1) - (x-1) & 0 \\ 1 & 1 & x-1 \end{vmatrix} = \begin{vmatrix} x-1 & 1 & 1 \\ 0 & 0 & 0 \\ 1 & 1 & x-1 \end{vmatrix} \]

### Step 3: Factor out common terms
Next, we notice that the second row is now all zeros, which means we can factor out \(x-2\) from the determinant:
\[ D = (x-2) \begin{vmatrix} x-1 & 1 \\ 1 & x-1 \end{vmatrix} \]

### Step 4: Calculate the smaller determinant
Now we calculate the smaller 2x2 determinant:
\[ \begin{vmatrix} x-1 & 1 \\ 1 & x-1 \end{vmatrix} = (x-1)(x-1) - (1)(1) = (x-1)^2 - 1 \]
This simplifies to:
\[ (x-1)^2 - 1 = x^2 - 2x + 1 - 1 = x^2 - 2x \]

### Step 5: Combine the results
Now we combine everything:
\[ D = (x-2)(x^2 - 2x) = (x-2)x(x-2) \]
This can be rewritten as:
\[ D = (x-2)^2(x) \]

### Step 6: Set the determinant to zero
To find the roots, we set the determinant equal to zero:
\[ (x-2)^2(x) = 0 \]

### Step 7: Solve for the roots
The roots of this equation are:
1. \(x - 2 = 0 \Rightarrow x = 2\) (with multiplicity 2)
2. \(x = 0\)

Thus, the roots of the equation are \(x = 2\) (double root) and \(x = 0\).

### Final Answer
The roots of the equation are \(x = 2\) (with multiplicity 2) and \(x = 0\).

---