If A is a square matrix of order 3 and det A = 5, then what is det `[(2A)^(-1)]` equal to ?

Options

A. `(1)/(10)`
B. `(2)/(5)`
C. `(8)/(5)`
D. `(1)/(40)`

Answer

To solve the problem, we need to find the determinant of the matrix \((2A)^{-1}\) given that \(\text{det}(A) = 5\) and \(A\) is a square matrix of order 3.

### Step-by-Step Solution:

1. **Understanding the Determinant of a Scalar Multiple of a Matrix**:
The determinant of a scalar multiple of a matrix can be expressed as:
\[ \text{det}(kA) = k^n \cdot \text{det}(A) \]
where \(k\) is a scalar, \(A\) is an \(n \times n\) matrix, and \(n\) is the order of the matrix.

2. **Applying the Formula**:
In our case, \(k = 2\) and \(n = 3\) (since \(A\) is a \(3 \times 3\) matrix). Therefore:
\[ \text{det}(2A) = 2^3 \cdot \text{det}(A) = 8 \cdot 5 = 40 \]

3. **Finding the Determinant of the Inverse**:
The determinant of the inverse of a matrix is given by:
\[ \text{det}(A^{-1}) = \frac{1}{\text{det}(A)} \]
Thus, for the matrix \(2A\):
\[ \text{det}((2A)^{-1}) = \frac{1}{\text{det}(2A)} = \frac{1}{40} \]

4. **Final Result**:
Therefore, the determinant of \((2A)^{-1}\) is:
\[ \text{det}((2A)^{-1}) = \frac{1}{40} \]

### Conclusion:
The final answer is:
\[ \text{det}((2A)^{-1}) = \frac{1}{40} \]