Find the value of $A^2$, where $A$ is a $2 \times 2$ matrix whose elements are given by $a_{ij} = \begin{cases} 1 & \text{if } i \neq j \\ 0 & \text{if } i = j \end{cases}$.

$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

The correct answer is Option (1) → $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ ##

$a_{ij} = \begin{cases} 1 & \text{when } i \neq j \\ 0 & \text{when } i = j \end{cases}$

$A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

$A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

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

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