If A is a 3x3 matrix and B is its adjoint matrix the determinant of B is 64 then determinant of A is

Options

A. 64
B. `pm 64`
C. `pm 8`
D. 18

Answer

To find the determinant of matrix A given that the determinant of its adjoint matrix B is 64, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We know that for a square matrix \( A \), the determinant of its adjoint matrix \( B \) (denoted as \( \text{adj}(A) \)) is related to the determinant of \( A \) by the formula:
\[ \text{det}(\text{adj}(A)) = (\text{det}(A))^{n-1} \]
where \( n \) is the order of the matrix. Since \( A \) is a \( 3 \times 3 \) matrix, we have \( n = 3 \).

2. **Applying the Formula**:
Therefore, we can write:
\[ \text{det}(B) = \text{det}(\text{adj}(A)) = (\text{det}(A))^{3-1} = (\text{det}(A))^2 \]

3. **Substituting the Given Value**:
We are given that \( \text{det}(B) = 64 \). Thus, we can set up the equation:
\[ (\text{det}(A))^2 = 64 \]

4. **Solving for \( \text{det}(A) \)**:
To find \( \text{det}(A) \), we take the square root of both sides:
\[ \text{det}(A) = \sqrt{64} \]
This gives us:
\[ \text{det}(A) = 8 \quad \text{or} \quad \text{det}(A) = -8 \]

5. **Final Answer**:
Therefore, the determinant of matrix \( A \) is:
\[ \text{det}(A) = \pm 8 \]