Let matrix `A=[{:(x,y,-z),(1,2,3),(1,1,2):}]` , where `x,y,z in N`. If `|adj(adj(adj(adjA)))|=4^(8)*5^(16)`, then the number of such `(x,y,z)` are

Options

A. `28`
B. `36`
C. `45`
D. `55`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understanding the Matrix and its Determinant
Given the matrix \( A = \begin{bmatrix} x & y & -z \\ 1 & 2 & 3 \\ 1 & 1 & 2 \end{bmatrix} \), we need to find the determinant of this matrix.

### Step 2: Calculate the Determinant of Matrix A
The determinant of a 3x3 matrix \( \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) is calculated using the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For our matrix \( A \):
\[ \text{det}(A) = x(2 \cdot 2 - 3 \cdot 1) - y(1 \cdot 2 - 3 \cdot 1) + (-z)(1 \cdot 1 - 2 \cdot 1) \]
Calculating the terms:
- \( 2 \cdot 2 - 3 \cdot 1 = 4 - 3 = 1 \)
- \( 1 \cdot 2 - 3 \cdot 1 = 2 - 3 = -1 \)
- \( 1 \cdot 1 - 2 \cdot 1 = 1 - 2 = -1 \)

Thus, we have:
\[ \text{det}(A) = x(1) - y(-1) - z(-1) = x + y + z \]

### Step 3: Setting Up the Equation
From the problem, we know that:
\[ | \text{adj(adj(adj(adj(A))))} | = 4^8 \cdot 5^{16} \]
Using the property of determinants of adjoint matrices, we have:
\[ | \text{adj}(A) | = |A|^{n-1} \]
where \( n \) is the order of the matrix (which is 3). Therefore:
\[ | \text{adj}(A) | = |A|^{2} \]
Continuing this process for four adjoints:
\[ | \text{adj(adj(adj(adj(A))))} | = |A|^{16} \]
Setting this equal to the given expression:
\[ |A|^{16} = 4^8 \cdot 5^{16} \]

### Step 4: Simplifying the Equation
We can rewrite \( 4^8 \) as \( (2^2)^8 = 2^{16} \):
\[ |A|^{16} = 2^{16} \cdot 5^{16} \]
Taking the 16th root of both sides:
\[ |A| = 2 \cdot 5 = 10 \]

### Step 5: Setting Up the Equation for x, y, z
We have:
\[ x + y + z = 10 \]
where \( x, y, z \) are natural numbers.

### Step 6: Finding the Number of Solutions
The number of natural number solutions to the equation \( x + y + z = 10 \) can be found using the stars and bars combinatorial method. The formula for the number of solutions in natural numbers is given by:
\[ \text{Number of solutions} = (n - 1) \choose (k - 1) \]
where \( n \) is the sum (10) and \( k \) is the number of variables (3):
\[ \text{Number of solutions} = (10 - 1) \choose (3 - 1) = 9 \choose 2 \]
Calculating \( 9 \choose 2 \):
\[ 9 \choose 2 = \frac{9 \cdot 8}{2 \cdot 1} = 36 \]

### Final Answer
Thus, the number of such \( (x, y, z) \) is **36**.