If matrix `A=[a_(ij)]_(2xx2^(,))` where `a_(ij)=1" if "i!=j`
`=0" if "i=j` then `A^(2)` is equal to :

Options

A. I
B. A
C. O
D. None of these.

Answer

To solve the problem, we need to find \( A^2 \) for the given matrix \( A \) defined as follows:

1. **Matrix Definition**: The matrix \( A \) is a \( 2 \times 2 \) matrix where:
- \( a_{ij} = 1 \) if \( i \neq j \)
- \( a_{ij} = 0 \) if \( i = j \)

2. **Constructing Matrix \( A \)**:
- For a \( 2 \times 2 \) matrix, we have:
\[ A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \]
- Since \( a_{11} = 0 \) (because \( i = j \)), \( a_{22} = 0 \) (because \( i = j \)), \( a_{12} = 1 \) (because \( i \neq j \)), and \( a_{21} = 1 \) (because \( i \neq j \)), we can write:
\[ A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

3. **Calculating \( A^2 \)**:
- To find \( A^2 \), we compute \( A \times A \):
\[ A^2 = A \times A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

4. **Matrix Multiplication**:
- We perform the multiplication step-by-step:
- First row, first column:
\[ 0 \times 0 + 1 \times 1 = 0 + 1 = 1 \]
- First row, second column:
\[ 0 \times 1 + 1 \times 0 = 0 + 0 = 0 \]
- Second row, first column:
\[ 1 \times 0 + 0 \times 1 = 0 + 0 = 0 \]
- Second row, second column:
\[ 1 \times 1 + 0 \times 0 = 1 + 0 = 1 \]

5. **Resulting Matrix**:
- Combining these results, we have:
\[ A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]
- This is the identity matrix \( I \).

6. **Conclusion**:
- Therefore, \( A^2 = I \).

### Final Answer:
\[ A^2 = I \]