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 step by step, we will analyze the given information and derive the necessary equations.

### Step 1: Understand the properties of the adjoint matrix
The determinant of the adjoint of a matrix \( A \) of order \( n \) is given by the formula:
\[ |\text{adj}(A)| = |A|^{n-1} \]
For the adjoint of the adjoint matrix \( \text{adj}(\text{adj}(A)) \), the determinant is given by:
\[ |\text{adj}(\text{adj}(A))| = |A|^{(n-1)(n-2)} \]

### Step 2: Set up the equation based on the problem statement
According to the problem, we have:
\[ |\text{adj}(\text{adj}(A))| = |A|^{2n^2 - 7n + 7} \]
Equating the two expressions for \( |\text{adj}(\text{adj}(A))| \):
\[ |A|^{(n-1)(n-2)} = |A|^{2n^2 - 7n + 7} \]

### Step 3: Compare the exponents
Since \( |A| \neq 1 \), we can equate the exponents:
\[ (n-1)(n-2) = 2n^2 - 7n + 7 \]

### Step 4: Expand and rearrange the equation
Expanding the left side:
\[ n^2 - 3n + 2 = 2n^2 - 7n + 7 \]
Rearranging gives:
\[ 0 = 2n^2 - n - 5n + 7 - 2 \]
\[ 0 = 2n^2 - 5n + 5 \]

### Step 5: Simplify the equation
Rearranging the equation:
\[ 2n^2 - 5n + 5 = 0 \]

### Step 6: Use the quadratic formula
To find the roots of the quadratic equation \( 2n^2 - 5n + 5 = 0 \), we use the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 2 \), \( b = -5 \), and \( c = 5 \):
\[ n = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 5}}{2 \cdot 2} \]
\[ n = \frac{5 \pm \sqrt{25 - 40}}{4} \]
\[ n = \frac{5 \pm \sqrt{-15}}{4} \]

### Step 7: Analyze the roots
Since the discriminant \( 25 - 40 = -15 \) is negative, there are no real roots for \( n \). This means there are no possible values of \( n \) that satisfy the original equation.

### Step 8: Conclusion
Since there are no real values of \( n \), the sum of all possible values of \( n \) is:
\[ \text{Sum of all possible values of } n = 0 \]