If A is square matrix, A' its transpose and `|A|=2`, then `|A A'|` is

Options

A. 2
B. 4
C. 8
D. 16

Answer

To solve the problem, we need to find the determinant of the product of a square matrix \( A \) and its transpose \( A' \). Given that \( |A| = 2 \), we can use properties of determinants to find \( |A A'| \).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We know that \( A \) is a square matrix and \( |A| = 2 \). The transpose of \( A \) is denoted as \( A' \).

**Hint**: Recall that the determinant of a matrix provides important information about the matrix, including its invertibility.

2. **Using the Property of Determinants**:
One important property of determinants is that the determinant of the transpose of a matrix is equal to the determinant of the matrix itself:
\[ |A'| = |A| \]

**Hint**: This property is true for any square matrix, so you can always use it in similar problems.

3. **Applying the Determinant Product Rule**:
Another property states that the determinant of the product of two matrices is the product of their determinants:
\[ |A A'| = |A| \cdot |A'| \]

**Hint**: This property is useful when dealing with products of matrices.

4. **Substituting the Known Values**:
Since we know \( |A| = 2 \) and \( |A'| = |A| = 2 \), we can substitute these values into the equation:
\[ |A A'| = |A| \cdot |A'| = 2 \cdot 2 \]

**Hint**: Make sure to carefully substitute the values you have for the determinants.

5. **Calculating the Result**:
Now, we can compute the product:
\[ |A A'| = 4 \]

**Hint**: Always double-check your arithmetic to ensure accuracy.

### Final Answer:
Thus, the determinant \( |A A'| \) is \( 4 \).