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

Options

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

Answer

To find \( A^2 \) for the given matrix \( A \), we will follow these steps:

### Step 1: Define the matrix \( A \)

The matrix \( A \) is defined as follows:
\[ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \]
where:
- \( a_{ij} = 1 \) if \( i \neq j \)
- \( a_{ij} = 0 \) if \( i = j \)

For a \( 2 \times 2 \) matrix, we can determine the elements:
- \( a_{11} = 0 \) (since \( i = j \))
- \( a_{12} = 1 \) (since \( i \neq j \))
- \( a_{21} = 1 \) (since \( i \neq j \))
- \( a_{22} = 0 \) (since \( i = j \))

Thus, the matrix \( A \) is:
\[ A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

### Step 2: Calculate \( A^2 \)

To find \( A^2 \), we multiply matrix \( A \) by itself:
\[ A^2 = A \cdot A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

### Step 3: Perform the matrix multiplication

We calculate the elements of \( A^2 \):

1. **First row, first column**:
\[ (0 \cdot 0) + (1 \cdot 1) = 0 + 1 = 1 \]

2. **First row, second column**:
\[ (0 \cdot 1) + (1 \cdot 0) = 0 + 0 = 0 \]

3. **Second row, first column**:
\[ (1 \cdot 0) + (0 \cdot 1) = 0 + 0 = 0 \]

4. **Second row, second column**:
\[ (1 \cdot 1) + (0 \cdot 0) = 1 + 0 = 1 \]

Putting these results together, we get:
\[ A^2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

### Step 4: Conclusion

The matrix \( A^2 \) is the identity matrix \( I \):
\[ A^2 = I \]