If A be any square matix of order `3xx3` and `|A|=5`, then find the value of `|adj(adjA)|`

Answer

To solve the problem of finding the value of \(|\text{adj}(\text{adj} A)|\) for a square matrix \(A\) of order \(3 \times 3\) where \(|A| = 5\), we can follow these steps:

### Step 1: Understand the properties of determinants and adjoints
The determinant of the adjoint of a matrix \(A\) of order \(n\) is given by the formula:
\[ |\text{adj}(A)| = |A|^{n-1} \]
where \(n\) is the order of the matrix.

### Step 2: Calculate \(|\text{adj}(A)|\)
Since \(A\) is a \(3 \times 3\) matrix, we have:
\[ n = 3 \]
Thus, we can calculate \(|\text{adj}(A)|\):
\[ |\text{adj}(A)| = |A|^{3-1} = |A|^2 \]
Given that \(|A| = 5\), we find:
\[ |\text{adj}(A)| = 5^2 = 25 \]

### Step 3: Calculate \(|\text{adj}(\text{adj} A)|\)
Now we need to find \(|\text{adj}(\text{adj} A)|\). Again using the property of determinants:
\[ |\text{adj}(\text{adj} A)| = |\text{adj}(A)|^{n-1} \]
Since \(|\text{adj}(A)| = 25\) and \(n = 3\), we have:
\[ |\text{adj}(\text{adj} A)| = |\text{adj}(A)|^{3-1} = |\text{adj}(A)|^2 \]
Substituting the value we found:
\[ |\text{adj}(\text{adj} A)| = 25^2 = 625 \]

### Final Answer
Thus, the value of \(|\text{adj}(\text{adj} A)|\) is:
\[ \boxed{625} \]