Let `A=[a_(ij)]_(mxxn)` be a matrix such that `a_(ij)=1` for all I,j. Then ,

Options

A. rank (A) gt 1
B. rank (A) = 1
C. rank (A) = m
D. rank (A) = n

Answer

To solve the problem, we need to analyze the given matrix \( A = [a_{ij}]_{m \times n} \) where each element \( a_{ij} = 1 \) for all \( i, j \).

### Step-by-step Solution:

1. **Define the Matrix**:
The matrix \( A \) is defined as:
\[ A = \begin{bmatrix} 1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1 \end{bmatrix} \]
This matrix has \( m \) rows and \( n \) columns, and every element is equal to 1.

**Hint**: Visualize the structure of the matrix to understand its uniformity.

2. **Identify the Rows and Columns**:
Since every element of the matrix is 1, all rows are identical, and all columns are identical.

**Hint**: Think about what it means for rows and columns to be identical in terms of linear dependence.

3. **Determine the Rank**:
The rank of a matrix is defined as the maximum number of linearly independent rows or columns. Since all rows are identical, they are linearly dependent. The same applies to the columns.

**Hint**: Recall that if all rows (or columns) are multiples of each other, the rank is determined by the number of unique rows or columns.

4. **Conclusion about the Rank**:
In this case, there is only one unique row (or column), which is the row (or column) of all ones. Therefore, the rank of the matrix \( A \) is 1.

**Hint**: Consider the implications of having only one unique row or column in terms of the rank.

### Final Answer:
The rank of the matrix \( A \) is 1.