A, B and C are three square matrices of order 3 such that A= diag `(x, y, z)`, det `(B)=4` and det `(C)=2`, where `x, y, z in I^(+)`. If det`(adj (adj (ABC)))``=2^(16)xx3^(8)xx7^(4)`, then the number of distinct possible matrices A is ________ .

Explanation

To solve the problem, we need to determine the number of distinct possible matrices \( A \) given the conditions provided. Let's break it down step by step.

### Step 1: Understanding the Matrices
We have three square matrices \( A, B, \) and \( C \) of order 3. The matrix \( A \) is a diagonal matrix given by:
\[ A = \text{diag}(x, y, z) \]
where \( x, y, z \) are positive integers.

### Step 2: Determinants of Matrices
We are given:
- \( \det(B) = 4 \)
- \( \det(C) = 2 \)

The determinant of matrix \( A \) can be calculated as:
\[ \det(A) = x \cdot y \cdot z \]

### Step 3: Determinant of the Product of Matrices
The determinant of the product of matrices is given by:
\[ \det(ABC) = \det(A) \cdot \det(B) \cdot \det(C) \]
Substituting the known values, we have:
\[ \det(ABC) = (xyz) \cdot 4 \cdot 2 = 8xyz \]

### Step 4: Determinant of the Adjoint
The determinant of the adjoint of a matrix \( M \) is given by:
\[ \det(\text{adj}(M)) = \det(M)^{n-1} \]
where \( n \) is the order of the matrix. For a 3x3 matrix, \( n = 3 \), so:
\[ \det(\text{adj}(M)) = \det(M)^2 \]
Thus, for the adjoint of the adjoint:
\[ \det(\text{adj}(\text{adj}(M))) = \det(M)^{(n-1)^2} = \det(M)^4 \]

Applying this to our case:
\[ \det(\text{adj}(\text{adj}(ABC))) = \det(ABC)^4 \]

### Step 5: Setting Up the Equation
We know from the problem statement that:
\[ \det(\text{adj}(\text{adj}(ABC))) = 2^{16} \cdot 3^8 \cdot 7^4 \]
Thus, we have:
\[ (8xyz)^4 = 2^{16} \cdot 3^8 \cdot 7^4 \]

### Step 6: Simplifying the Equation
Calculating \( (8xyz)^4 \):
\[ 8 = 2^3 \implies (8xyz)^4 = (2^3xyz)^4 = 2^{12} \cdot (xyz)^4 \]
So we can equate:
\[ 2^{12} \cdot (xyz)^4 = 2^{16} \cdot 3^8 \cdot 7^4 \]

### Step 7: Isolating \( xyz \)
From the equation, we can isolate \( (xyz)^4 \):
\[ (xyz)^4 = 2^{16-12} \cdot 3^8 \cdot 7^4 = 2^4 \cdot 3^8 \cdot 7^4 \]
Taking the fourth root:
\[ xyz = 2^{1} \cdot 3^{2} \cdot 7^{1} = 2 \cdot 9 \cdot 7 = 126 \]

### Step 8: Finding Distinct Values of \( x, y, z \)
Now we need to find the distinct positive integer combinations of \( x, y, z \) such that:
\[ xyz = 126 \]
The prime factorization of \( 126 \) is:
\[ 126 = 2^1 \cdot 3^2 \cdot 7^1 \]

### Step 9: Distributing Factors
We can distribute the factors \( 2^1, 3^2, 7^1 \) among \( x, y, z \). The distinct combinations can be found by considering the partitions of the factors.

1. **Distribution of \( 1, 14 \)**:
- Possible combinations: \( (1, 1, 14), (1, 2, 7) \)
- The arrangements of \( (1, 1, 14) \) gives \( \frac{3!}{2!} = 3 \).
- The arrangements of \( (1, 2, 7) \) gives \( 3! = 6 \).

### Step 10: Total Combinations
Adding the distinct arrangements:
\[ 3 + 6 = 9 \]

Thus, the number of distinct possible matrices \( A \) is:
\[ \boxed{9} \]