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 Order of the Matrix**:
Since \(A\) is a square matrix of order \(3\), we denote this order as \(N = 3\).

2. **Using the Property of Adjoint**:
The determinant of the adjoint of a matrix \(A\) can be expressed using the formula:
\[ |Adj(A)| = |A|^{N-1} \]
where \(N\) is the order of the matrix.

3. **Scaling the Matrix**:
We need to find the adjoint of \(4A\). Using the property of determinants, we have:
\[ |Adj(kA)| = k^{N-1} |Adj(A)| \]
where \(k\) is a scalar. In our case, \(k = 4\).

4. **Calculating the Adjoint of \(4A\)**:
Therefore, we can express the determinant of the adjoint of \(4A\) as:
\[ |Adj(4A)| = |4A|^{N-1} \]
First, we need to calculate \(|4A|\):
\[ |4A| = 4^N |A| = 4^3 \cdot 5 = 64 \cdot 5 = 320 \]

5. **Finding the Determinant of the Adjoint**:
Now, substituting back into our formula for the adjoint:
\[ |Adj(4A)| = |4A|^{N-1} = (320)^{3-1} = (320)^2 \]

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

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