Answer:
We have, \[A=\left[ \begin{matrix} x & 0 \\ 1 & 1 \\ \end{matrix} \right]\]and \[B=\left[ \begin{matrix} 1 & 0 \\ 5 & 1 \\ \end{matrix} \right]\] Now, \[{{A}^{2}}=\left[ \begin{matrix} x & 0 \\ 1 & 1 \\ \end{matrix} \right]\,\,\left[ \begin{matrix} x & 0 \\ 1 & 1 \\ \end{matrix} \right]\] \[=\left[ \begin{matrix} {{x}^{2}}+0 & 0+0 \\ x+1 & 0+1 \\ \end{matrix} \right]=\left[ \begin{matrix} {{x}^{2}} & 0 \\ x+1 & 1 \\ \end{matrix} \right]\] As, \[{{A}^{2}}=B\] [given] \[\left[ \begin{matrix} {{x}^{2}} & 0 \\ x+1 & 1 \\ \end{matrix} \right]=\left[ \begin{matrix} 1 & 0 \\ 5 & 1 \\ \end{matrix} \right]\] On equating both sides, we get \[{{x}^{2}}=1\Rightarrow x=\pm \,\,1\]and \[x+1=5\Rightarrow x=4\] As, there is no unique value of x. Thus, no value of x exists.
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