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

Options

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

Explanation

To solve the equation given by the determinant

\[ \begin{vmatrix} 1 & t-1 & 1 \\ t-1 & 1 & 1 \\ 1 & 1 & t-1 \end{vmatrix} = 0, \]

we will compute the determinant step by step and find the roots.

### Step 1: Calculate the Determinant

We will use the formula for the determinant of a 3x3 matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:

\[ A = \begin{pmatrix} 1 & t-1 & 1 \\ t-1 & 1 & 1 \\ 1 & 1 & t-1 \end{pmatrix} \]

where \(a = 1\), \(b = t-1\), \(c = 1\), \(d = t-1\), \(e = 1\), \(f = 1\), \(g = 1\), \(h = 1\), \(i = t-1\).

Calculating the determinant:

\[ \text{det}(A) = 1 \cdot (1 \cdot (t-1) - 1 \cdot 1) - (t-1) \cdot ((t-1) \cdot (t-1) - 1 \cdot 1) + 1 \cdot ((t-1) \cdot 1 - 1 \cdot 1) \]

### Step 2: Simplify Each Term

1. The first term simplifies to:
\[ 1 \cdot ((t-1) - 1) = t - 2 \]

2. The second term simplifies to:
\[ -(t-1) \cdot ((t-1)^2 - 1) = -(t-1) \cdot (t^2 - 2t + 1 - 1) = -(t-1)(t^2 - 2t) = -(t-1)t(t-2) \]

3. The third term simplifies to:
\[ 1 \cdot ((t-1) - 1) = t - 2 \]

### Step 3: Combine the Terms

Now, we combine all the terms:

\[ \text{det}(A) = (t - 2) - (t - 1)t(t - 2) + (t - 2) \]

This simplifies to:

\[ \text{det}(A) = 2(t - 2) - (t - 1)t(t - 2) \]

### Step 4: Set the Determinant to Zero

Now we set the determinant to zero:

\[ 2(t - 2) - (t - 1)t(t - 2) = 0 \]

Factoring out \(t - 2\):

\[ (t - 2)(2 - (t - 1)t) = 0 \]

### Step 5: Solve for Roots

This gives us two cases to solve:

1. \(t - 2 = 0 \Rightarrow t = 2\)
2. \(2 - (t - 1)t = 0\)

For the second case:

\[ (t - 1)t = 2 \Rightarrow t^2 - t - 2 = 0 \]

Factoring gives:

\[ (t - 2)(t + 1) = 0 \]

Thus, the roots are:

\[ t = 2 \quad \text{and} \quad t = -1 \]

### Final Roots

The roots of the equation are:

\[ t = 2 \quad \text{and} \quad t = -1 \]

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