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

Explanation

To solve the problem, we need to understand the relationship between a determinant and its cofactor matrix. Given that \( D \) is a determinant of order 3, we denote it as:

\[ D = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} \]

The cofactor matrix of \( D \) is obtained by replacing each element of \( D \) with its corresponding cofactor. The determinant of this cofactor matrix is denoted as \( D' \).

### Step-by-Step Solution:

1. **Understanding the Cofactor Matrix**:
The cofactor \( C_{ij} \) of an element \( a_{ij} \) in a matrix is given by \( (-1)^{i+j} \) times the determinant of the submatrix formed by deleting the \( i \)-th row and \( j \)-th column from the original matrix.

2. **Determinant of the Cofactor Matrix**:
For a determinant of order \( n \), the relationship between the determinant \( D \) and the determinant of its cofactor matrix \( D' \) is given by the formula:
\[ D' = D^{n-1} \]
where \( n \) is the order of the determinant.

3. **Applying the Formula**:
Since \( D \) is a determinant of order 3, we have \( n = 3 \). Therefore, we can substitute \( n \) into the formula:
\[ D' = D^{3-1} = D^2 \]

4. **Conclusion**:
Thus, the determinant \( D' \) obtained by replacing the elements of \( D \) by their cofactors is equal to \( D^2 \).

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