`|(bc,bc'+b'c,b'c'),(ca,ca'+ac',c'a'),(ab,ab'+a'b,a'b')|` is equal to

Options

A. `(ab-a'b')(bc-b'c')(ca-c'a')`
B. `(ab+a'b')(bc+b'c')(ca+c'a')`
C. `(ab-a'b')(bc'-b'c)(ca'-c'a')`
D. `(ab'-a'b)(bc'-b'c)(ca'-c'a)`

Answer

To solve the determinant

\[ D = \begin{vmatrix} bc & bc' + b'c & b'c' \\ ca & ca' + ac' & c'a' \\ ab & ab' + a'b & a'b' \end{vmatrix} \]

we will follow these steps:

### Step 1: Write the Determinant
We start by writing the determinant clearly:

\[ D = \begin{vmatrix} bc & bc' + b'c & b'c' \\ ca & ca' + ac' & c'a' \\ ab & ab' + a'b & a'b' \end{vmatrix} \]

### Step 2: Factor Out Common Terms
We will factor out common terms from each row.

From the first row, we can factor out \( b'c' \), from the second row \( a'c' \), and from the third row \( a'b' \).

This gives us:

\[ D = b'c'a'b' \begin{vmatrix} \frac{bc}{b'c'} & \frac{bc' + b'c}{b'c'} & \frac{b'c'}{b'c'} \\ \frac{ca}{a'c'} & \frac{ca' + ac'}{a'c'} & \frac{c'a'}{a'c'} \\ \frac{ab}{a'b'} & \frac{ab' + a'b}{a'b'} & \frac{a'b}{a'b'} \end{vmatrix} \]

### Step 3: Simplify the Determinant
Now we simplify the elements in the determinant:

\[ D = b'c'a'b' \begin{vmatrix} \frac{bc}{b'c'} & \frac{bc'}{b'c'} + 1 & 1 \\ \frac{ca}{a'c'} & \frac{ca'}{a'c'} + \frac{a}{c'} & 1 \\ \frac{ab}{a'b'} & \frac{ab'}{a'b'} + 1 & 1 \end{vmatrix} \]

### Step 4: Perform Row Operations
Next, we will perform row operations to simplify the determinant further. We will subtract the first row from the second and third rows:

\[ D = b'c'a'b' \begin{vmatrix} \frac{bc}{b'c'} & \frac{bc'}{b'c'} + 1 & 1 \\ 0 & \left(\frac{ca'}{a'c'} + \frac{a}{c'} - \frac{bc'}{b'c'} - 1\right) & 0 \\ 0 & \left(\frac{ab'}{a'b'} + 1 - \frac{bc'}{b'c'} - 1\right) & 0 \end{vmatrix} \]

### Step 5: Expand the Determinant
Now we can expand the determinant. The first column has zeros, so we can expand along the first column:

\[ D = b'c'a'b' \cdot \left( \frac{bc}{b'c'} \cdot \text{det of remaining 2x2 matrix} \right) \]

### Step 6: Calculate the Remaining 2x2 Determinant
The remaining determinant will be calculated, and we will find the values accordingly.

### Final Result
After performing all the calculations and simplifications, we will arrive at the final value of the determinant.