The value of the determinant `|(-1,1,1),(1,-1,1),(1,1,-1)|` is equal to

Options

A. `- 4`
B. 0
C. 1
D. 4

Answer

To find the value of the determinant \( D = \begin{vmatrix} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{vmatrix} \), we will use the method of cofactor expansion along the first row.

### Step 1: Write down the determinant
\[ D = \begin{vmatrix} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{vmatrix} \]

### Step 2: Expand the determinant along the first row
Using the formula for the determinant, we expand it as follows:
\[ D = -1 \cdot \begin{vmatrix} -1 & 1 \\ 1 & -1 \end{vmatrix} - 1 \cdot \begin{vmatrix} 1 & 1 \\ 1 & -1 \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & -1 \\ 1 & 1 \end{vmatrix} \]

### Step 3: Calculate the 2x2 determinants
1. Calculate \( \begin{vmatrix} -1 & 1 \\ 1 & -1 \end{vmatrix} \):
\[ = (-1)(-1) - (1)(1) = 1 - 1 = 0 \]

2. Calculate \( \begin{vmatrix} 1 & 1 \\ 1 & -1 \end{vmatrix} \):
\[ = (1)(-1) - (1)(1) = -1 - 1 = -2 \]

3. Calculate \( \begin{vmatrix} 1 & -1 \\ 1 & 1 \end{vmatrix} \):
\[ = (1)(1) - (-1)(1) = 1 + 1 = 2 \]

### Step 4: Substitute back into the determinant expression
Now substituting these values back into the expression for \( D \):
\[ D = -1 \cdot 0 - 1 \cdot (-2) + 1 \cdot 2 \]
\[ D = 0 + 2 + 2 = 4 \]

### Final Answer
The value of the determinant is \( D = 4 \).

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