The value of `|{:(a- b, b- c, c - a),(b-c,c-a,a-b),(c-a,a-b,b-c):}|` =

Options

A. 0
B. `-1`
C. 1
D. 2

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} a - b & b - c & c - a \\ b - c & c - a & a - b \\ c - a & a - b & b - c \end{vmatrix} \]

we will follow these steps:

### Step 1: Write down the determinant
We start with the determinant as given:

\[ D = \begin{vmatrix} a - b & b - c & c - a \\ b - c & c - a & a - b \\ c - a & a - b & b - c \end{vmatrix} \]

### Step 2: Apply column operations
We will apply the column operation \( C_1 \leftarrow C_1 + C_2 + C_3 \). This means we will add the second and third columns to the first column.

Calculating the new first column:

- First row: \( (a - b) + (b - c) + (c - a) = a - b + b - c + c - a = 0 \)
- Second row: \( (b - c) + (c - a) + (a - b) = b - c + c - a + a - b = 0 \)
- Third row: \( (c - a) + (a - b) + (b - c) = c - a + a - b + b - c = 0 \)

So the new determinant becomes:

\[ D = \begin{vmatrix} 0 & b - c & c - a \\ 0 & c - a & a - b \\ 0 & a - b & b - c \end{vmatrix} \]

### Step 3: Identify the row of zeros
Now, we can see that the first column consists entirely of zeros.

### Step 4: Conclude the value of the determinant
Since any determinant with a row (or column) of zeros is equal to zero, we conclude that:

\[ D = 0 \]

Thus, the value of the determinant is

\[ \boxed{0} \]