If A is an invertible square matrix of the order n such that `|A| ne1` and `|adj(adjA)|=|A|^((2n^(2)-7n+7))` then the sum of all possible values of n is

Options

A. 3
B. 4
C. 6
D. 5

Answer

To solve the problem, we need to analyze the given condition involving the determinant of the adjoint of a matrix \( A \).

1. **Understanding the adjoint matrix**: The adjoint of a matrix \( A \) is denoted as \( \text{adj}(A) \). For any square matrix \( A \) of order \( n \), the determinant of the adjoint matrix is given by the formula:
\[ |\text{adj}(A)| = |A|^{n-1} \]

2. **Finding the adjoint of the adjoint**: We need to find \( |\text{adj}(\text{adj}(A))| \). Using the property of determinants:
\[ |\text{adj}(\text{adj}(A))| = |\text{adj}(A)|^{n-1} \]
Substituting the earlier result, we get:
\[ |\text{adj}(\text{adj}(A))| = (|A|^{n-1})^{n-1} = |A|^{(n-1)(n-1)} = |A|^{(n-1)^2} \]

3. **Setting up the equation**: According to the problem, we have:
\[ |\text{adj}(\text{adj}(A))| = |A|^{2n^2 - 7n + 7} \]
Therefore, we can equate the two expressions we derived:
\[ |A|^{(n-1)^2} = |A|^{2n^2 - 7n + 7} \]

4. **Comparing the exponents**: Since \( |A| \neq 0 \) (as \( A \) is invertible), we can equate the exponents:
\[ (n-1)^2 = 2n^2 - 7n + 7 \]

5. **Expanding and rearranging**: Expanding the left side:
\[ n^2 - 2n + 1 = 2n^2 - 7n + 7 \]
Rearranging gives:
\[ 0 = 2n^2 - 7n + 7 - n^2 + 2n - 1 \]
Simplifying this leads to:
\[ 0 = n^2 - 5n + 6 \]

6. **Factoring the quadratic**: The quadratic can be factored as:
\[ (n-2)(n-3) = 0 \]
Thus, the solutions for \( n \) are:
\[ n = 2 \quad \text{or} \quad n = 3 \]

7. **Finding the sum of all possible values of \( n \)**: The sum of all possible values of \( n \) is:
\[ 2 + 3 = 5 \]

Thus, the final answer is:
\[ \boxed{5} \]