If matrix 𝐴 = $\begin{bmatrix$}[r] 2 & 2 \ 0 & 2 \end{bmatrix}\text{ and } A^2 = \begin{bmatrix}[r] 4 & x \ 0 & 4 \end{bmatrix}, then the value of x is :

1. 2

2. 4

3. 8

4. 10

$\phantom{\Rightarrow} A^2 = \begin{bmatrix$}[r] 2 & 2 \ 0 & 2 \end{bmatrix}\begin{bmatrix}[r] 2 & 2 \ 0 & 2 \end{bmatrix} \\[1em] \Rightarrow A^2 = \begin{bmatrix}[r] 2 \times 2 + 2\times 0 & 2 \times 2 + 2 \times 2 \ 0 \times 2 + 2 \times 0 & 0 \times 2 + 2 \times 2 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 4 & x \ 0 & 4 \end{bmatrix} = \begin{bmatrix}[r] 4 + 0 & 4 + 4 \ 0 + 0 & 0 + 4 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 4 & x \ 0 & 4 \end{bmatrix} = \begin{bmatrix}[r] 4 & 8 \ 0 & 4 \end{bmatrix} \\[1em] \Rightarrow x = 8.

Hence, Option 3 is the correct option.