Let $\mathrm{A}=\left[\begin{array}{ll}\mathrm{x} & 1 \\ 1 & 0\end{array}\right], \mathrm{x} \in \mathrm{R}$ and $\mathrm{A}^{4}=\left[\mathrm{a}_{\mathrm{ij}}\right] .$ If
$a_{11}=109$, then $a_{22}$ is equal to____________
$A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]$
$A^{2}=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{cc}x^{2}+1 & x \\ x & 1\end{array}\right]$
$A^{4}=\left[\begin{array}{cc}x^{2}+1 & x \\ x & 1\end{array}\right]\left[\begin{array}{cc}x^{2}+1 & x \\ x & 1\end{array}\right]$
$=\left[\begin{array}{cc}\left(x^{2}+1\right)^{2}+x^{2} & x\left(x^{2}+1\right)+x \\ x\left(x^{2}+1\right)+x & x^{2}+1\end{array}\right]$
$a_{11}=\left(x^{2}+1\right)^{2}+x^{2}=109$
$\Rightarrow x=\pm 3$
$a_{22}=x^{2}+1=10$
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