The value of determinant `|{:(a-b,b+c,a),(b-a,c+a,b),(c-a,a+b,c):}|` is

Options

A. `a^(3)+b^(3)+c^(3)`
B. `3bc`
C. `a^(3)+b^(2)+c^(3)-3abc`
D. None of these

Answer

To find the value of the determinant

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

we will perform column operations to simplify the determinant.

### Step 1: Perform Column Operation
Let’s perform the operation \( C_1 \to C_3 - C_1 \). This means we will subtract the first column from the third column.

\[ C_1 = \begin{pmatrix} a-b \\ b-a \\ c-a \end{pmatrix}, \quad C_3 = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \]

After the operation, the new third column becomes:

\[ C_3' = C_3 - C_1 = \begin{pmatrix} a - (a-b) \\ b - (b-a) \\ c - (c-a) \end{pmatrix} = \begin{pmatrix} b \\ a \\ a \end{pmatrix} \]

So the determinant now looks like:

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

### Step 2: Perform Another Column Operation
Next, we will perform the operation \( C_2 \to C_2 - C_1 \).

The new second column becomes:

\[ C_2' = C_2 - C_1 = \begin{pmatrix} (b+c) - (a-b) \\ (c+a) - (b-a) \\ (a+b) - (c-a) \end{pmatrix} = \begin{pmatrix} b+c - a + b \\ c+a - b + a \\ a+b - c + a \end{pmatrix} = \begin{pmatrix} 2b + c - a \\ 2a + c - b \\ 2a + b - c \end{pmatrix} \]

So the determinant now looks like:

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

### Step 3: Expand the Determinant
Now we can expand the determinant using the first column:

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

### Step 4: Calculate the 2x2 Determinants
Calculating each of the 2x2 determinants will yield values based on the entries of the matrix.

After performing the calculations, we will find that the determinant simplifies to:

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

### Final Result
After simplifying, we can conclude that the value of the determinant is:

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