If A is a square matrix of order 3 such that `|A^(T)| = 5`, then value of |2A|= a) 25 b) 10 c) 20 d) 40

Options

A. 25
B. 10
C. 20
D. 30

Answer

To solve the problem, we need to find the value of \(|2A|\) given that \(|A^T| = 5\) and \(A\) is a square matrix of order 3.

### Step-by-step solution:

1. **Understanding the Determinant of Transpose**:
We know that for any square matrix \(A\), the determinant of the transpose of \(A\) is equal to the determinant of \(A\):
\[ |A^T| = |A| \]
Given that \(|A^T| = 5\), we can conclude:
\[ |A| = 5 \]

**Hint**: Remember that the determinant of a matrix and its transpose are always equal.

2. **Finding the Determinant of a Scalar Multiple of a Matrix**:
The determinant of a scalar multiple of a matrix can be calculated using the formula:
\[ |kA| = k^n |A| \]
where \(k\) is a scalar, \(n\) is the order of the matrix, and \(|A|\) is the determinant of the matrix \(A\).

3. **Applying the Formula**:
In our case, we want to find \(|2A|\) where \(k = 2\) and \(n = 3\) (since \(A\) is a \(3 \times 3\) matrix):
\[ |2A| = 2^3 |A| \]
Now, substituting the value of \(|A|\):
\[ |2A| = 2^3 \cdot 5 \]

4. **Calculating the Value**:
Calculate \(2^3\):
\[ 2^3 = 8 \]
Now, substituting back:
\[ |2A| = 8 \cdot 5 = 40 \]

5. **Final Answer**:
Therefore, the value of \(|2A|\) is:
\[ \boxed{40} \]