If `A` is a square matrix of order `3` such that `|A|=5`, then `|Adj(4A)|=`

Options

A. `5^(3)xx4^(2)`
B. `5^(2)xx4^(3)`
C. `5^(2)xx16^(3)`
D. `5^(3)xx16^(2)`

Explanation

To solve the problem, we need to find the determinant of the adjoint of the matrix \(4A\) given that the determinant of matrix \(A\) is \(5\) and \(A\) is a square matrix of order \(3\).

### Step-by-step Solution:

1. **Understanding the adjoint of a scalar multiple of a matrix**:
The formula for the adjoint of a scalar multiple of a matrix is given by:
\[ \text{Adj}(kA) = k^{n-1} \cdot \text{Adj}(A) \]
where \(n\) is the order of the matrix. In this case, \(k = 4\) and \(n = 3\).

2. **Applying the formula**:
Using the formula, we can express the adjoint of \(4A\):
\[ \text{Adj}(4A) = 4^{3-1} \cdot \text{Adj}(A) = 4^2 \cdot \text{Adj}(A) = 16 \cdot \text{Adj}(A) \]

3. **Finding the determinant of the adjoint**:
The determinant of the adjoint of a matrix \(A\) is given by:
\[ |\text{Adj}(A)| = |A|^{n-1} \]
For our matrix \(A\), since \(|A| = 5\) and \(n = 3\):
\[ |\text{Adj}(A)| = |A|^{3-1} = |A|^2 = 5^2 = 25 \]

4. **Calculating the determinant of \(4A\)**:
The determinant of a scalar multiple of a matrix is given by:
\[ |kA| = k^n \cdot |A| \]
Therefore, for \(4A\):
\[ |4A| = 4^3 \cdot |A| = 64 \cdot 5 = 320 \]

5. **Finding the determinant of the adjoint of \(4A\)**:
Now, we can find the determinant of the adjoint of \(4A\) using the previously mentioned property:
\[ |\text{Adj}(4A)| = |4A|^{n-1} = |4A|^{3-1} = |4A|^2 \]
Substituting the value of \(|4A|\):
\[ |\text{Adj}(4A)| = 320^2 \]

6. **Calculating \(320^2\)**:
\[ 320^2 = 102400 \]

### Final Answer:
Thus, the value of \(|\text{Adj}(4A)|\) is \(102400\).