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

Options

A. I
B. A
C. Zero matrix
D. None of these

Answer

To solve the problem, we need to find the square of the matrix \( A \), which is defined as follows:

Given:
\[ A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

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

From the problem statement, we know that:
- \( a_{ij} = 0 \) if \( i = j \)
- \( a_{ij} = 1 \) if \( i \neq j \)

For a \( 2 \times 2 \) matrix, we can explicitly write:
\[ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]

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

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

### Step 3: Perform the matrix multiplication

We will calculate each element of the resulting matrix:

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

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

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

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

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

### Final Result

Thus, the square of the matrix \( A \) is:
\[ A^2 = I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]