If `A` is a square matrix such that `A(adjA)=[(4,0,0),(0,4,0),(0,0,4)],` then `=(|adj(adjA)|)/(2|adjA|)` is equal to

Explanation

To solve the problem step-by-step, we will follow the reasoning presented in the video transcript.

### Step 1: Understand the given information
We are given that:
\[ A \cdot \text{adj}(A) = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{pmatrix} \]
This matrix can be factored as:
\[ A \cdot \text{adj}(A) = 4I \]
where \( I \) is the identity matrix.

### Step 2: Use the property of determinants
From the property of determinants, we know:
\[ A \cdot \text{adj}(A) = \det(A) \cdot I \]
By comparing this with the previous step, we can conclude:
\[ \det(A) = 4 \]

### Step 3: Find the determinant of adjoint of A
Using the formula for the determinant of the adjoint of a matrix:
\[ \det(\text{adj}(A)) = \det(A)^{n-1} \]
where \( n \) is the order of the matrix \( A \). Since \( A \) is a \( 3 \times 3 \) matrix, \( n = 3 \):
\[ \det(\text{adj}(A)) = \det(A)^{3-1} = \det(A)^2 = 4^2 = 16 \]

### Step 4: Find the determinant of adjoint of adjoint of A
Now, we need to find \( \det(\text{adj}(\text{adj}(A))) \). Using the same property:
\[ \det(\text{adj}(B)) = \det(B)^{m-1} \]
where \( m \) is the order of the matrix \( B \). Here, \( B = \text{adj}(A) \), which is also a \( 3 \times 3 \) matrix. Therefore:
\[ \det(\text{adj}(\text{adj}(A))) = \det(\text{adj}(A))^{3-1} = \det(\text{adj}(A))^2 = 16^2 = 256 \]

### Step 5: Calculate the final expression
We need to find:
\[ \frac{\det(\text{adj}(\text{adj}(A)))}{2 \cdot \det(\text{adj}(A))} \]
Substituting the values we found:
\[ \frac{256}{2 \cdot 16} = \frac{256}{32} = 8 \]

### Final Answer
Thus, the value is:
\[ \boxed{8} \]