if `A_(1) ,B_(1),C_(1) …….` are respectively the cofactors of the elements `a_(1) ,b_(1),c_(1)……` of the determinant
`Delta = |{:(a_(1),,b_(1),,c_(1)),(a_(2),,b_(2),,c_(2)),(a_(3),,b_(3),,c_(3)):}|, Delta ne 0 ` then the value of `|{:(B_(2),,C_(2)),(B_(3),,C_(3)):}|` is equal to

Options

A. `a_(1)^(2)Delta`
B. `a_(1)Delta`
C. `a_(1)Delta^(2)`
D. `a_(1)^(2)Delta^(2)`

Explanation

To solve the given problem, we need to find the value of the determinant \( |{:(B_2, C_2),(B_3, C_3):}| \) where \( B_2, C_2, B_3, C_3 \) are the cofactors of the elements \( b_2, c_2, b_3, c_3 \) respectively from the determinant \( \Delta = |{:(a_1, b_1, c_1),(a_2, b_2, c_2),(a_3, b_3, c_3):}| \).

### Step 1: Write the determinant
The determinant is given as:
\[ \Delta = |{:(a_1, b_1, c_1),(a_2, b_2, c_2),(a_3, b_3, c_3):}| \]

### Step 2: Calculate the cofactors
The cofactors \( B_2, C_2, B_3, C_3 \) can be calculated as follows:

- **Cofactor \( B_2 \)**:
\[ B_2 = (-1)^{2+1} |{:(a_1, c_1),(a_3, c_3):}| = |{:(a_1, c_1),(a_3, c_3):}| = a_1 c_3 - a_3 c_1 \]

- **Cofactor \( C_2 \)**:
\[ C_2 = (-1)^{2+2} |{:(a_1, b_1),(a_3, b_3):}| = |{:(a_1, b_1),(a_3, b_3):}| = a_1 b_3 - a_3 b_1 \]

- **Cofactor \( B_3 \)**:
\[ B_3 = (-1)^{3+2} |{:(a_1, c_1),(a_2, c_2):}| = -|{:(a_1, c_1),(a_2, c_2):}| = - (a_1 c_2 - a_2 c_1) = a_2 c_1 - a_1 c_2 \]

- **Cofactor \( C_3 \)**:
\[ C_3 = (-1)^{3+3} |{:(a_1, b_1),(a_2, b_2):}| = |{:(a_1, b_1),(a_2, b_2):}| = a_1 b_2 - a_2 b_1 \]

### Step 3: Set up the determinant of cofactors
Now we need to find the determinant:
\[ |{:(B_2, C_2),(B_3, C_3):}| \]
Substituting the values of \( B_2, C_2, B_3, C_3 \):
\[ = |{:(a_1 c_3 - a_3 c_1, a_1 b_3 - a_3 b_1),(a_2 c_1 - a_1 c_2, a_1 b_2 - a_2 b_1):}| \]

### Step 4: Calculate the determinant
Using the determinant formula for a 2x2 matrix:
\[ |{:(x_1, y_1),(x_2, y_2):}| = x_1 y_2 - x_2 y_1 \]
we can compute:
\[ = (a_1 c_3 - a_3 c_1)(a_1 b_2 - a_2 b_1) - (a_2 c_1 - a_1 c_2)(a_1 b_3 - a_3 b_1) \]

### Step 5: Simplify the expression
After expanding and simplifying the above expression, we will get:
\[ = a_1^2 (c_3 b_2 - c_1 b_3) + a_2 (c_1 b_3 - c_3 b_1) + a_3 (c_2 b_1 - c_1 b_2) \]

### Conclusion
Thus, the final value of the determinant \( |{:(B_2, C_2),(B_3, C_3):}| \) is:
\[ = \Delta \]
where \( \Delta \) is the original determinant.