If D is determinant of order 3 and D' is the determinant obtained by replacing the elements of D by their cofactors, then which one of the following is correct ?

Options

A. A) `D' = D^(2)`
B. B) `D' = D^(3)`
C. C) `D' = 2D^(2)`
D. D) `D' = 3D^(3)`

Answer

To solve the problem, we need to establish the relationship between the determinant \( D \) of order 3 and the determinant \( D' \) obtained by replacing the elements of \( D \) with their cofactors.

### Step-by-Step Solution:

1. **Understanding the Determinant and Cofactors**:
- A determinant \( D \) of order 3 is a 3x3 matrix. The cofactor of an element in a matrix is calculated by taking the determinant of the submatrix that remains after removing the row and column of that element, multiplied by \((-1)^{i+j}\), where \( i \) and \( j \) are the row and column indices of the element.

2. **Cofactor Matrix**:
- When we replace each element of the determinant \( D \) with its corresponding cofactor, we form a new matrix known as the cofactor matrix. The determinant of this cofactor matrix is denoted as \( D' \).

3. **Property of Determinants**:
- There is a known property regarding the determinant of the cofactor matrix. For a determinant of order \( n \), the determinant of the cofactor matrix \( D' \) is related to the original determinant \( D \) by the formula:
\[ D' = D^{n-1} \]

4. **Applying the Property**:
- Since we are given that the order \( n = 3 \), we can substitute this value into the property:
\[ D' = D^{3-1} = D^2 \]

5. **Conclusion**:
- Therefore, the relationship between the determinant \( D \) and the determinant \( D' \) obtained by replacing the elements of \( D \) with their cofactors is:
\[ D' = D^2 \]

### Final Answer:
The correct relationship is \( D' = D^2 \).