For a `3xx3` matrix A if det A=4, then det (Adj. A) equals

Options

A. `-4`
B. `4`
C. `16`
D. `64`

Answer

To find the determinant of the adjoint of a \(3 \times 3\) matrix \(A\) given that \(\text{det}(A) = 4\), we can use the following formula:

\[ \text{det}(\text{Adj} A) = (\text{det} A)^{n-1} \]

where \(n\) is the order of the matrix. In this case, since \(A\) is a \(3 \times 3\) matrix, \(n = 3\).

### Step-by-Step Solution:

1. **Identify the order of the matrix**:
\[ n = 3 \]

2. **Use the formula for the determinant of the adjoint**:
\[ \text{det}(\text{Adj} A) = (\text{det} A)^{n-1} \]

3. **Substitute the given value of \(\text{det}(A)\)**:
\[ \text{det}(\text{Adj} A) = (4)^{3-1} \]

4. **Calculate \(3 - 1\)**:
\[ 3 - 1 = 2 \]

5. **Now calculate \(4^2\)**:
\[ 4^2 = 16 \]

6. **Conclusion**:
\[ \text{det}(\text{Adj} A) = 16 \]

### Final Answer:
\[ \text{det}(\text{Adj} A) = 16 \]