A is a square matrix of order 3 such that its determinant is 4. What is the determinant of its transpose ?

Options

A. A) 64
B. B) 36
C. C) 32
D. D) 4

Answer

To solve the problem, we need to find the determinant of the transpose of a square matrix \( A \) of order 3, given that the determinant of \( A \) is 4.

### Step-by-step solution:

1. **Understanding the properties of determinants**:
We start by recalling a fundamental property of determinants: the determinant of a matrix is equal to the determinant of its transpose. Mathematically, this can be expressed as:
\[ \text{det}(A) = \text{det}(A^T) \]

2. **Given information**:
We are given that the determinant of matrix \( A \) is:
\[ \text{det}(A) = 4 \]

3. **Applying the property**:
Using the property from step 1, we can write:
\[ \text{det}(A^T) = \text{det}(A) \]

4. **Substituting the known value**:
Since we know that \(\text{det}(A) = 4\), we can substitute this value into our equation:
\[ \text{det}(A^T) = 4 \]

5. **Conclusion**:
Therefore, the determinant of the transpose of matrix \( A \) is:
\[ \text{det}(A^T) = 4 \]

### Final Answer:
The determinant of the transpose of matrix \( A \) is \( 4 \).

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