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 \( k = |A \cdot \text{adj} A| \) for a scalar matrix \( A \) whose elements are the roots of the polynomial equation \( x^9 - 15x^8 + 75x^7 - 125x^6 = 0 \).

### Step 1: Factor the Polynomial
We start by factoring the polynomial equation given:

\[ x^9 - 15x^8 + 75x^7 - 125x^6 = 0 \]

We can factor out \( x^6 \):

\[ x^6(x^3 - 15x^2 + 75x - 125) = 0 \]

This gives us \( x = 0 \) as a root with multiplicity 6. Now we need to find the roots of the cubic polynomial:

\[ x^3 - 15x^2 + 75x - 125 = 0 \]

### Step 2: Find Roots of the Cubic Polynomial
Using the Rational Root Theorem or synthetic division, we can test for possible rational roots. Testing \( x = 5 \):

\[ 5^3 - 15 \cdot 5^2 + 75 \cdot 5 - 125 = 125 - 375 + 375 - 125 = 0 \]

Thus, \( x = 5 \) is a root. We can factor the cubic polynomial as:

\[ (x - 5)(x^2 - 10x + 25) = 0 \]

The quadratic \( x^2 - 10x + 25 \) can be factored further:

\[ (x - 5)^2 = 0 \]

Thus, \( x = 5 \) is a root with multiplicity 2. Therefore, the roots of the original polynomial are:

- \( x = 0 \) (multiplicity 6)
- \( x = 5 \) (multiplicity 3)

### Step 3: Form the Scalar Matrix \( A \)
Since \( A \) is a scalar matrix, all diagonal elements are equal to the same scalar value. The scalar value will be the most repeated root, which is \( 5 \). Therefore, we can represent \( A \) as:

\[ A = \begin{pmatrix} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{pmatrix} \]

### Step 4: Calculate \( |A| \)
The determinant of a scalar matrix \( A \) is given by:

\[ |A| = 5^3 = 125 \]

### Step 5: Calculate \( |\text{adj} A| \)
The adjugate of a matrix \( A \) is related to the determinant by the formula:

\[ |\text{adj} A| = |A|^{n-1} \]

where \( n \) is the order of the matrix. Here \( n = 3 \):

\[ |\text{adj} A| = |A|^{3-1} = |A|^2 = 125^2 = 15625 \]

### Step 6: Calculate \( |A \cdot \text{adj} A| \)
Now, we can find \( |A \cdot \text{adj} A| \):

\[ |A \cdot \text{adj} A| = |A| \cdot |\text{adj} A| = 125 \cdot 15625 \]

Calculating this gives:

\[ 125 \cdot 15625 = 1953125 \]

Thus, the value of \( k \) is:

\[ \boxed{1953125} \]