Let A be a matrix of rank r. Then,

Options

A. `"rank " (A^T)=r`
B. `"rank "(A^T)ltr`
C. `"rank " (A^T)gtr`
D. none of these

Answer

To solve the problem, we need to determine the rank of the transpose of a matrix \( A \) given that the rank of \( A \) is \( r \).

### Step-by-Step Solution:

1. **Understanding Rank**:
The rank of a matrix is defined as the maximum number of linearly independent row vectors or column vectors in the matrix. If the rank of matrix \( A \) is \( r \), it means there are \( r \) linearly independent rows or columns in \( A \).

**Hint**: Recall that the rank indicates the number of non-zero rows or columns in the matrix.

2. **Transpose of a Matrix**:
The transpose of a matrix \( A \), denoted as \( A^T \), is formed by swapping the rows and columns of \( A \). Thus, if \( A \) has dimensions \( m \times n \), then \( A^T \) will have dimensions \( n \times m \).

**Hint**: Remember that transposing a matrix does not change the linear independence of its rows or columns; it merely interchanges them.

3. **Rank of the Transpose**:
It is a fundamental property of matrices that the rank of a matrix is equal to the rank of its transpose. Therefore, if the rank of \( A \) is \( r \), then the rank of \( A^T \) is also \( r \).

**Hint**: Use the property that \( \text{rank}(A) = \text{rank}(A^T) \) to conclude the result.

4. **Conclusion**:
Since we have established that the rank of \( A^T \) is equal to the rank of \( A \), we can conclude that the rank of \( A^T \) is also \( r \).

**Hint**: Summarize your findings by stating that the rank remains unchanged under transposition.

### Final Answer:
The rank of \( A^T \) is \( r \).