If `A = [a_(ij)]` is a square matrix of order 2 such that `a_(ij)={{:(1,"when " i ne j),(0,"when "i = j):}` that `A^2` is :

Options

A. `[(1,0),(1,0)]`
B. `[(1,1),(0,0)]`
C. `[(1,1),(1,0)]`
D. `[(1,0),(0,1)]`

Answer

To find \( A^2 \) for the given matrix \( A \), let's first define the matrix based on the provided conditions.

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

Given that \( a_{ij} = 1 \) when \( i \neq j \) and \( a_{ij} = 0 \) when \( i = j \), for a square matrix of order 2, we can write:

\[ A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

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

To find \( A^2 \), we need to multiply the matrix \( A \) by itself:

\[ A^2 = A \times A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \times \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \]

### Step 3: Perform the matrix multiplication

We will perform the multiplication step by step:

- **Element at (1,1)**:
\[ 0 \cdot 0 + 1 \cdot 1 = 0 + 1 = 1 \]

- **Element at (1,2)**:
\[ 0 \cdot 1 + 1 \cdot 0 = 0 + 0 = 0 \]

- **Element at (2,1)**:
\[ 1 \cdot 0 + 0 \cdot 1 = 0 + 0 = 0 \]

- **Element at (2,2)**:
\[ 1 \cdot 1 + 0 \cdot 0 = 1 + 0 = 1 \]

### Step 4: Write the resulting matrix

Combining all the results, we get:

\[ A^2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]

### Step 5: Identify the answer

The resulting matrix \( A^2 \) is the identity matrix, which corresponds to option 4:

\[ \text{Option 4: } \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]