Let `A=[a_(ij)]_(3xx3)` be a scalar matrix whose elements are the roots of the equation `x^(9)-15x^(8)+75x^(7)-125x^(6)=0`. If `|A.adjA|=k`, then the vlaue of k is equal to

Options

A. `5^(12)`
B. `5^(9)`
C. `3^(12)`
D. `3^(9)`

Answer

To solve the problem, we need to find the value of \( k \) where \( |A \cdot \text{adj}(A)| = k \) for a scalar matrix \( A \) whose elements are the roots of the equation \( x^9 - 15x^8 + 75x^7 - 125x^6 = 0 \).

### Step 1: Factor the polynomial
The given polynomial can be factored by taking out \( x^6 \):
\[ x^9 - 15x^8 + 75x^7 - 125x^6 = x^6(x^3 - 15x^2 + 75x - 125) = 0 \]
This shows that \( x = 0 \) is a root with multiplicity 6.

### Step 2: Find the roots of the cubic equation
Now we need to solve the cubic equation:
\[ x^3 - 15x^2 + 75x - 125 = 0 \]
Using Vieta's formulas, the sum of the roots \( r_1 + r_2 + r_3 = 15 \) and the product \( r_1 r_2 r_3 = 125 \).

### Step 3: Find the roots of the cubic equation
We can find the roots of the cubic equation using synthetic division or by trial and error. Testing \( x = 5 \):
\[ 5^3 - 15 \cdot 5^2 + 75 \cdot 5 - 125 = 125 - 375 + 375 - 125 = 0 \]
So, \( x = 5 \) is a root. We can factor the cubic as:
\[ (x - 5)(x^2 - 10x + 25) = 0 \]
The quadratic \( x^2 - 10x + 25 = (x - 5)^2 \) gives us another double root. Thus, the roots of the polynomial are \( 0 \) (with multiplicity 6) and \( 5 \) (with multiplicity 3).

### Step 4: Construct the scalar matrix \( A \)
Since \( A \) is a scalar matrix, we can represent it as:
\[ A = \begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix} \]

### Step 5: Calculate \( |A| \)
The determinant of \( A \) is:
\[ |A| = 5^3 = 125 \]

### Step 6: Use the property of adjoint
The property of the adjoint states that:
\[ |\text{adj}(A)| = |A|^{n-1} \]
where \( n \) is the order of the matrix. For a \( 3 \times 3 \) matrix, \( n = 3 \):
\[ |\text{adj}(A)| = |A|^{3-1} = |A|^2 = 125^2 = 15625 \]

### Step 7: Calculate \( |A \cdot \text{adj}(A)| \)
Now, we calculate:
\[ |A \cdot \text{adj}(A)| = |A| \cdot |\text{adj}(A)| = 125 \cdot 15625 \]
Calculating this gives:
\[ 125 \cdot 15625 = 5^3 \cdot (5^2)^3 = 5^3 \cdot 5^6 = 5^{9} \]

### Conclusion
Thus, we find that:
\[ k = 5^9 \]