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 determine the rank of the matrix \( A = [a_{ij}]_{m \times n} \) where \( a_{ij} = 1 \) for all \( i, j \).

### Step-by-step Solution:

1. **Understanding the Matrix**:
The matrix \( A \) is an \( m \times n \) matrix where every element is equal to 1. This means that all rows of the matrix are identical.

2. **Identifying the Rows**:
Since every row of the matrix is the same (i.e., each row is \( [1, 1, \ldots, 1] \)), we can see that the rows are linearly dependent.

3. **Row Operations**:
To find the rank, we can perform row operations. For instance, we can subtract one row from another. If we take the first row and subtract the second row, we will get a row of zeros:
\[ R_1 - R_2 \rightarrow R_1 \]
This operation will yield:
\[ [1, 1, \ldots, 1] - [1, 1, \ldots, 1] = [0, 0, \ldots, 0] \]
Repeating this for all pairs of rows will result in all rows except one being transformed into rows of zeros.

4. **Resulting Matrix**:
After performing the row operations, the matrix will look like this:
\[ \begin{bmatrix} 1 & 1 & \ldots & 1 \\ 0 & 0 & \ldots & 0 \\ 0 & 0 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & 0 \end{bmatrix} \]
This matrix has only one non-zero row.

5. **Determining the Rank**:
The rank of a matrix is defined as the maximum number of linearly independent row vectors in the matrix. Since we have transformed the matrix such that there is only one non-zero row, the rank of the matrix \( A \) is 1.

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