The value of `|(b +c,a,a),(b,c +a,b),(c,c,a +b)|`, is

Options

A. 6 abc
B. `a +b +c`
C. 4 abc
D. abc

Answer

To find the value of the determinant

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

we will use the method of cofactor expansion.

### Step 1: Write the determinant
We start by writing the determinant explicitly as follows:

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

### Step 2: Expand the determinant
We will expand the determinant along the first row:

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

### Step 3: Calculate the 2x2 determinants
Now we calculate each of the 2x2 determinants:

1. For the first determinant:

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

2. For the second determinant:

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

3. For the third determinant:

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

### Step 4: Substitute back into the expansion
Now substituting back into our expression for \(D\):

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

### Step 5: Simplify the expression
Expanding each term:

1. Expanding \((b+c)(ac + ab + a^2)\):

\[ = bac + bab + ba^2 + cac + cab + ca^2 = abc + ab^2 + a^2b + ac^2 + abc + a^2c \]

2. Expanding \(-a(ab + b^2 - bc)\):

\[ = -a^2b - ab^2 + abc \]

3. Expanding \(a(bc - c^2 - ac)\):

\[ = abc - ac^2 - a^2c \]

### Step 6: Combine all terms
Now we combine all terms:

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

### Step 7: Collect like terms
Combining like terms:

- The \(abc\) terms: \(3abc\)
- The \(ab^2\) terms: \(ab^2 - ab^2 = 0\)
- The \(a^2b\) terms: \(a^2b - a^2b = 0\)
- The \(ac^2\) terms: \(ac^2 - ac^2 = 0\)
- The \(a^2c\) terms: \(a^2c - a^2c = 0\)

Thus, we find that:

\[ D = 3abc \]

### Final Answer
The value of the determinant is

\[ \boxed{3abc} \]