If `A=[(0,-1,-4),(1,0,-7),(4,7,0)]` then `A^(T)=`

Options

A. `O`
B. I
C. A
D. `-A`

Answer

To find the transpose of the matrix \( A \), we will follow these steps:

1. **Identify the matrix \( A \)**:
\[ A = \begin{pmatrix} 0 & -1 & -4 \\ 1 & 0 & -7 \\ 4 & 7 & 0 \end{pmatrix} \]

2. **Understand the concept of transpose**:
The transpose of a matrix is obtained by swapping its rows with columns. This means that the first row of \( A \) becomes the first column of \( A^T \), the second row becomes the second column, and the third row becomes the third column.

3. **Apply the transpose operation**:
- The first row of \( A \) is \( (0, -1, -4) \), which will become the first column of \( A^T \).
- The second row of \( A \) is \( (1, 0, -7) \), which will become the second column of \( A^T \).
- The third row of \( A \) is \( (4, 7, 0) \), which will become the third column of \( A^T \).

Therefore, we can write:
\[ A^T = \begin{pmatrix} 0 & 1 & 4 \\ -1 & 0 & 7 \\ -4 & -7 & 0 \end{pmatrix} \]

4. **Final result**:
Thus, the transpose of the matrix \( A \) is:
\[ A^T = \begin{pmatrix} 0 & 1 & 4 \\ -1 & 0 & 7 \\ -4 & -7 & 0 \end{pmatrix} \]