If matrix 𝐴 = [2202] and A2=[4x04]\begin{bmatrix*}[r] 2 & 2 \\ 0 & 2 \end{bmatrix*}\text{ and } A^2 = \begin{bmatrix*}[r] 4 & x \\ 0 & 4 \end{bmatrix*}[2022] and A2=[40x4], then the value of x is :
2
4
8
10
⇒A2=[2202][2202]⇒A2=[2×2+2×02×2+2×20×2+2×00×2+2×2]⇒[4x04]=[4+04+40+00+4]⇒[4x04]=[4804]⇒x=8.\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.⇒A2=[2022][2022]⇒A2=[2×2+2×00×2+2×02×2+2×20×2+2×2]⇒[40x4]=[4+00+04+40+4]⇒[40x4]=[4084]⇒x=8.
Hence, Option 3 is the correct option.
