The value of determinant A of 3rd order is 9 then the value of `Delta'^(2)` where `Delta'` is a determinant formed by cofactors of the element of `Delta` is

Options

A. 9
B. 81
C. 729
D. 6561

Answer

To solve the problem, we need to find the value of \( \Delta'^2 \), where \( \Delta' \) is the determinant formed by the cofactors of a given determinant \( \Delta \) of order 3, which has a value of 9.

### Step-by-Step Solution:

1. **Understanding the Determinant of Cofactors**:
The determinant formed by the cofactors of a matrix \( A \) (denoted as \( \Delta' \)) has a specific relationship with the determinant of the original matrix \( A \). The relationship is given by:
\[ \Delta' = \text{det}(A)^{n-1} \]
where \( n \) is the order of the matrix. In this case, since \( A \) is a 3rd order matrix, \( n = 3 \).

2. **Calculating \( \Delta' \)**:
Given that \( \text{det}(A) = 9 \), we can substitute this into the formula:
\[ \Delta' = \text{det}(A)^{3-1} = \text{det}(A)^{2} = 9^{2} \]

3. **Calculating \( 9^{2} \)**:
Now, we calculate \( 9^{2} \):
\[ 9^{2} = 81 \]
Therefore, \( \Delta' = 81 \).

4. **Finding \( \Delta'^2 \)**:
Finally, we need to find \( \Delta'^2 \):
\[ \Delta'^2 = (81)^{2} \]

5. **Calculating \( (81)^{2} \)**:
Now we calculate \( (81)^{2} \):
\[ (81)^{2} = 6561 \]

### Final Result:
Thus, the value of \( \Delta'^2 \) is \( 6561 \).

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