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 matrices A is equal to (where, `|M|` represents determinant of a matrix M)

Answer

To solve the problem, we need to find the number of matrices \( A \) of the form

\[ A = \begin{pmatrix} x & y & -z \\ 1 & 2 & 3 \\ 1 & 1 & 2 \end{pmatrix} \]

where \( x, y, z \in \mathbb{N} \) (natural numbers) and it is given that

\[ |\text{adj}(\text{adj}(\text{adj}(\text{adj}(A))))| = 4^8 \cdot 5^{16}. \]

### Step 1: Understanding the adjoint and determinant relationship

For a \( 3 \times 3 \) matrix \( A \), the relationship between the determinant of \( A \) and its adjoint is given by:

\[ |\text{adj}(A)| = |A|^{n-1} = |A|^2 \quad \text{(since \( n = 3 \))} \]

### Step 2: Calculate the determinant of the adjoint

Continuing this process, we find:

\[ |\text{adj}(\text{adj}(A))| = |\text{adj}(A)|^{n-1} = |A|^2 \cdot |A|^2 = |A|^4 \]

\[ |\text{adj}(\text{adj}(\text{adj}(A)))| = |\text{adj}(\text{adj}(A))|^{n-1} = |A|^4 \cdot |A|^2 = |A|^6 \]

\[ |\text{adj}(\text{adj}(\text{adj}(\text{adj}(A))))| = |\text{adj}(\text{adj}(\text{adj}(A)))|^{n-1} = |A|^6 \cdot |A|^2 = |A|^8 \]

### Step 3: Set up the equation

From the problem, we have:

\[ |A|^8 = 4^8 \cdot 5^{16} \]

### Step 4: Simplify the right-hand side

We can express \( 4^8 \) as \( (2^2)^8 = 2^{16} \), thus:

\[ |A|^8 = 2^{16} \cdot 5^{16} \]

Taking the eighth root of both sides gives:

\[ |A| = 2^2 \cdot 5^2 = 4 \cdot 25 = 100 \]

### Step 5: Calculate the determinant of matrix \( A \)

Next, we compute the determinant of 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, we find:

\[ |A| = x \begin{vmatrix} 2 & 3 \\ 1 & 2 \end{vmatrix} - y \begin{vmatrix} 1 & 3 \\ 1 & 2 \end{vmatrix} - z \begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} \]

Calculating the smaller determinants:

\[ \begin{vmatrix} 2 & 3 \\ 1 & 2 \end{vmatrix} = (2 \cdot 2 - 3 \cdot 1) = 4 - 3 = 1 \]

\[ \begin{vmatrix} 1 & 3 \\ 1 & 2 \end{vmatrix} = (1 \cdot 2 - 3 \cdot 1) = 2 - 3 = -1 \]

\[ \begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} = (1 \cdot 1 - 2 \cdot 1) = 1 - 2 = -1 \]

Thus, we have:

\[ |A| = x(1) - y(-1) - z(-1) = x + y + z \]

### Step 6: Set the determinant equal to 100

We set the equation:

\[ x + y + z = 100 \]

### Step 7: Finding the number of solutions

To find the number of natural number solutions to \( x + y + z = 100 \), we can use the stars and bars combinatorial method. The formula for the number of solutions in natural numbers is:

\[ \text{Number of solutions} = \binom{n-1}{k-1} \]

where \( n \) is the total (100) and \( k \) is the number of variables (3). Thus:

\[ \text{Number of solutions} = \binom{100-1}{3-1} = \binom{99}{2} \]

Calculating \( \binom{99}{2} \):

\[ \binom{99}{2} = \frac{99 \times 98}{2} = 4851 \]

### Final Answer

The number of such matrices \( A \) is equal to \( 4851 \).

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