If `D=|(a^(2)+1,ab,ac),(ba,b^(2)+1,bc),(ca,cb,c^(2)+1)|` then D equal to

Options

A. `1+a^(2)+b^(2)+c^(2)`
B. `a^(2)b^(2)c^(2)`
C. `bc+ca+ab`
D. zero

Answer

To solve the determinant \( D = \begin{vmatrix} a^2 + 1 & ab & ac \\ ba & b^2 + 1 & bc \\ ca & cb & c^2 + 1 \end{vmatrix} \), we will perform a series of row operations and simplifications.

### Step-by-Step Solution:

1. **Multiply Rows by Variables**:
- Multiply Row 1 by \( a \), Row 2 by \( b \), and Row 3 by \( c \):
\[ D = \begin{vmatrix} a(a^2 + 1) & a(ab) & a(ac) \\ b(ba) & b(b^2 + 1) & b(bc) \\ c(ca) & c(cb) & c(c^2 + 1) \end{vmatrix} \]
This gives:
\[ D = \begin{vmatrix} a^3 + a & a^2b & a^2c \\ b^2a & b^3 + b & b^2c \\ c^2a & c^2b & c^3 + c \end{vmatrix} \]

2. **Factor Out \( abc \)**:
- Since we multiplied each row, we need to divide by \( abc \):
\[ D = \frac{1}{abc} \begin{vmatrix} a^3 + a & a^2b & a^2c \\ b^2a & b^3 + b & b^2c \\ c^2a & c^2b & c^3 + c \end{vmatrix} \]

3. **Simplify the Determinant**:
- We can factor out \( a \), \( b \), and \( c \) from each respective row:
\[ D = \frac{1}{abc} \cdot abc \begin{vmatrix} a^2 + 1 & ab & ac \\ ba & b^2 + 1 & bc \\ ca & cb & c^2 + 1 \end{vmatrix} \]
This simplifies to:
\[ D = \begin{vmatrix} a^2 + 1 & ab & ac \\ ba & b^2 + 1 & bc \\ ca & cb & c^2 + 1 \end{vmatrix} \]

4. **Row Operation**:
- Perform the row operation \( R_1 \to R_1 + R_2 + R_3 \):
\[ D = \begin{vmatrix} (a^2 + 1) + (b^2 + 1) + (c^2 + 1) & ab + ba + cb & ac + bc + ca \\ ba & b^2 + 1 & bc \\ ca & cb & c^2 + 1 \end{vmatrix} \]
Which gives:
\[ D = \begin{vmatrix} a^2 + b^2 + c^2 + 3 & ab + ba + cb & ac + bc + ca \\ ba & b^2 + 1 & bc \\ ca & cb & c^2 + 1 \end{vmatrix} \]

5. **Factor Out Common Terms**:
- Factor out \( 1 + a^2 + b^2 + c^2 \):
\[ D = (1 + a^2 + b^2 + c^2) \begin{vmatrix} 1 & ab & ac \\ ba & 1 & bc \\ ca & cb & 1 \end{vmatrix} \]

6. **Simplify the Remaining Determinant**:
- Now, we can compute the determinant:
\[ D = (1 + a^2 + b^2 + c^2) \cdot 1 \]

7. **Final Result**:
- Thus, the value of \( D \) is:
\[ D = 1 + a^2 + b^2 + c^2 \]

### Final Answer:
\[ D = 1 + a^2 + b^2 + c^2 \]