If`A=[a_(ij)]_(mxxn)` is a matrix of rank r and B is a square submatrix of order r + 1, then

Options

A. B is invertible
B. B is not invertible
C. B many or may be invertible
D. none of these

Answer

To solve the problem, we need to analyze the properties of the matrix \( A \) and its submatrix \( B \).

### Step-by-Step Solution:

1. **Understanding the Matrix \( A \)**:
- Let \( A = [a_{ij}]_{m \times n} \) be a matrix of rank \( r \).
- This means that the maximum number of linearly independent rows or columns in \( A \) is \( r \).

**Hint**: Recall that the rank of a matrix indicates the number of linearly independent rows or columns.

2. **Submatrix \( B \)**:
- Let \( B \) be a square submatrix of order \( r + 1 \). This means \( B \) has dimensions \( (r + 1) \times (r + 1) \).

**Hint**: A square matrix of order \( r + 1 \) has \( r + 1 \) rows and \( r + 1 \) columns.

3. **Rank Condition**:
- Since the rank of \( A \) is \( r \), it cannot have more than \( r \) linearly independent rows or columns. Therefore, any square submatrix of order greater than \( r \) must be linearly dependent.

**Hint**: A matrix can have at most as many linearly independent rows (or columns) as its rank.

4. **Determinant of Submatrix \( B \)**:
- Since \( B \) is of order \( r + 1 \) and the rank of \( A \) is \( r \), the determinant of \( B \) must be zero. This is because a square matrix is invertible if and only if its determinant is non-zero.

**Hint**: A determinant of zero indicates that the rows (or columns) of the matrix are linearly dependent.

5. **Invertibility of Matrix \( B \)**:
- Since the determinant of \( B \) is zero, \( B \) is not invertible. A matrix is invertible if it has a non-zero determinant.

**Hint**: Remember that a matrix is invertible if it has a non-zero determinant.

### Conclusion:
From the steps above, we conclude that if \( A \) is a matrix of rank \( r \) and \( B \) is a square submatrix of order \( r + 1 \), then the determinant of \( B \) is zero, and thus \( B \) is not invertible.

### Final Answer:
The matrix \( B \) is not invertible.