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 need to find the values of \( x, y, z \) in the matrix \( A = \begin{pmatrix} x & y & -z \\ 1 & 2 & 3 \\ 1 & 1 & 2 \end{pmatrix} \) such that \( |adj(adj(adj(adj A)))| = 4^8 \cdot 5^{16} \).

### Step-by-Step Solution:

1. **Understanding the Determinant of the Adjoint**:
The determinant of the adjoint of a matrix \( A \) is given by the formula:
\[ |adj(A)| = |A|^{n-1} \]
where \( n \) is the order of the matrix. Here, \( A \) is a \( 3 \times 3 \) matrix, so \( n = 3 \).

2. **Finding the Determinant of the Adjoint**:
Therefore, the determinant of the adjoint of \( A \) is:
\[ |adj(A)| = |A|^{2} \]
For the adjoint of the adjoint:
\[ |adj(adj(A))| = |A|^{2} \cdot |A|^{2} = |A|^{4} \]
Continuing this process:
\[ |adj(adj(adj(A)))| = |A|^{8} \]
\[ |adj(adj(adj(adj(A))))| = |A|^{16} \]

3. **Setting Up the Equation**:
We know from the problem statement that:
\[ |adj(adj(adj(adj A)))| = 4^8 \cdot 5^{16} \]
Thus, we have:
\[ |A|^{16} = 4^8 \cdot 5^{16} \]

4. **Finding the Determinant of A**:
Taking the 16th root on both sides gives:
\[ |A| = (4^8 \cdot 5^{16})^{1/16} = 4^{1/2} \cdot 5 = 2 \cdot 5 = 10 \]

5. **Calculating the Determinant of A**:
Now we need to calculate the determinant of the matrix \( A \):
\[ |A| = \begin{vmatrix} x & y & -z \\ 1 & 2 & 3 \\ 1 & 1 & 2 \end{vmatrix} \]
Using the determinant formula for a \( 3 \times 3 \) matrix:
\[ |A| = x \begin{vmatrix} 2 & 3 \\ 1 & 2 \end{vmatrix} - y \begin{vmatrix} 1 & 3 \\ 1 & 2 \end{vmatrix} - z \begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} \]

6. **Calculating the Minors**:
The minors are calculated as follows:
\[ \begin{vmatrix} 2 & 3 \\ 1 & 2 \end{vmatrix} = (2 \cdot 2) - (3 \cdot 1) = 4 - 3 = 1 \]
\[ \begin{vmatrix} 1 & 3 \\ 1 & 2 \end{vmatrix} = (1 \cdot 2) - (3 \cdot 1) = 2 - 3 = -1 \]
\[ \begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} = (1 \cdot 1) - (2 \cdot 1) = 1 - 2 = -1 \]

7. **Substituting Back**:
Thus, we have:
\[ |A| = x \cdot 1 - y \cdot (-1) - z \cdot (-1) = x + y + z \]
Setting this equal to 10:
\[ x + y + z = 10 \]

8. **Finding Natural Number Solutions**:
Since \( x, y, z \) are natural numbers (i.e., positive integers), we can use the stars and bars combinatorial method to find the number of solutions. The number of solutions to \( x + y + z = 10 \) in natural numbers is given by:
\[ \text{Number of solutions} = \binom{10 - 1}{3 - 1} = \binom{9}{2} = \frac{9 \cdot 8}{2 \cdot 1} = 36 \]

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