If `A=[a_(ij)]_(mxxn)` is a matrix of rank r then (A) `r=min{m,n}` (B) `rlemin{m,n}` (C) `rltmin{m,n}` (D) none of these

Options

A. `r=min(m,n)`
B. `rltmin(m,n)`
C. `r lt min(m,n)`
D. none of these

Answer

To solve the problem, we need to analyze the rank of a matrix \( A \) which is defined as \( A = [a_{ij}]_{m \times n} \). The rank of a matrix is the maximum number of linearly independent row or column vectors in the matrix.

### Step-by-Step Solution:

1. **Understanding the Matrix Dimensions**:
- The matrix \( A \) has \( m \) rows and \( n \) columns.

2. **Definition of Rank**:
- The rank \( r \) of the matrix \( A \) is defined as the maximum number of linearly independent rows or columns.

3. **Rank Limitations**:
- The rank of a matrix cannot exceed the number of its rows or the number of its columns. Therefore, we have:
\[ r \leq \min(m, n) \]
- This means that the rank can be at most the smaller of the two dimensions \( m \) and \( n \).

4. **Possible Values of Rank**:
- The rank can be equal to \( \min(m, n) \) if all rows or columns are linearly independent. However, it can also be less than \( \min(m, n \) if there are dependencies among the rows or columns.

5. **Evaluating the Options**:
- (A) \( r = \min(m, n) \): This is not always true because the rank can be less than \( \min(m, n) \).
- (B) \( r \leq \min(m, n) \): This is true as explained above.
- (C) \( r < \min(m, n) \): This is not always true since the rank can also be equal to \( \min(m, n) \).
- (D) None of these: This is not applicable since option (B) is correct.

6. **Conclusion**:
- The correct answer is (B) \( r \leq \min(m, n) \).