the value of the determinant `|(b+c,a-b,a),(c+a,b-c,b),(a+b,c-a,c)|` is

Options

A. `a^3+b^3+c^3 -3abc `
B. `3abc -a^3-b^3-c^3`
C. `3abc +a^3+b^3+c^3`
D. none of these

Answer

To find the value of the determinant

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

we will perform a series of row operations and then calculate the determinant.

### Step 1: Row Operation
We will perform the operation \( R_1 \rightarrow R_1 + R_2 + R_3 \).

Calculating the new elements of \( R_1 \):
- First element: \( (b+c) + (c+a) + (a+b) = 2a + 2b + 2c \)
- Second element: \( (a-b) + (b-c) + (c-a) = 0 \)
- Third element: \( a + b + c \)

Thus, the new determinant becomes:

\[ D = \begin{vmatrix} 2(a+b+c) & 0 & a+b+c \\ c+a & b-c & b \\ a+b & c-a & c \end{vmatrix} \]

### Step 2: Factor out common terms
We can factor out \( a+b+c \) from the first row:

\[ D = (a+b+c) \begin{vmatrix} 2 & 0 & 1 \\ c+a & b-c & b \\ a+b & c-a & c \end{vmatrix} \]

### Step 3: Calculate the determinant
Now we will calculate the determinant of the remaining 3x3 matrix:

\[ D' = \begin{vmatrix} 2 & 0 & 1 \\ c+a & b-c & b \\ a+b & c-a & c \end{vmatrix} \]

Using the first row for expansion:

\[ D' = 2 \begin{vmatrix} b-c & b \\ c-a & c \end{vmatrix} - 0 + 1 \begin{vmatrix} c+a & b-c \\ a+b & c-a \end{vmatrix} \]

Calculating the first determinant:

\[ \begin{vmatrix} b-c & b \\ c-a & c \end{vmatrix} = (b-c)c - b(c-a) = bc - c^2 - bc + ab = ab - c^2 \]

Calculating the second determinant:

\[ \begin{vmatrix} c+a & b-c \\ a+b & c-a \end{vmatrix} = (c+a)(c-a) - (b-c)(a+b) = c^2 - a^2 + ab + bc - ac - b^2 + bc \]

Thus, we get:

\[ D' = 2(ab - c^2) + (c^2 - a^2 + ab + 2bc - b^2 - ac) \]

### Step 4: Combine terms
Now we combine the terms:

\[ D' = 2ab - 2c^2 + c^2 - a^2 + ab + 2bc - b^2 - ac \]
\[ = 3ab - a^2 - b^2 - ac - c^2 \]

### Step 5: Final determinant value
Now substituting back into \( D \):

\[ D = (a+b+c)(3ab - a^2 - b^2 - ac - c^2) \]

### Final Answer
Thus, the value of the determinant is:

\[ D = (a+b+c)(a^2 + b^2 + c^2 - ab - ac - bc) \]