Let $$A = \begin{bmatrix} x & 1 \\ 1 & 0 \end{bmatrix}$$, $$x \in R$$ and $$A^4 = [a_{ij}]$$. If $$a_{11} = 109$$, then $$a_{22}$$ is equal to __________

We have the matrix $$$A=\begin{bmatrix}x & 1\\ 1 & 0\end{bmatrix}\,.$$$

First we find $$A^2$$. Using the rule “to multiply two matrices, we take the dot-product of rows with columns”, we obtain

$$$A^2=A\cdot A=\begin{bmatrix}x & 1\\ 1 & 0\end{bmatrix} \begin{bmatrix}x & 1\\ 1 & 0\end{bmatrix}= \begin{bmatrix} x\cdot x+1\cdot 1 & x\cdot 1+1\cdot 0\\ 1\cdot x+0\cdot 1 & 1\cdot 1+0\cdot 0 \end{bmatrix} =\begin{bmatrix} x^{2}+1 & x\\ x & 1 \end{bmatrix}.$$$

Now we calculate $$A^3=A^2\cdot A$$:

$$$A^3=\begin{bmatrix}x^{2}+1 & x\\ x & 1\end{bmatrix} \begin{bmatrix}x & 1\\ 1 & 0\end{bmatrix} =\begin{bmatrix} (x^{2}+1)x+x\cdot 1 & (x^{2}+1)\cdot 1 + x\cdot 0\\[4pt] x\cdot x +1\cdot 1 & x\cdot 1 +1\cdot 0 \end{bmatrix} =\begin{bmatrix} x^{3}+2x & x^{2}+1\\ x^{2}+1 & x \end{bmatrix}.$$$

Next we form $$A^4=A^3\cdot A$$:

$$$A^4=\begin{bmatrix}x^{3}+2x & x^{2}+1\\ x^{2}+1 & x\end{bmatrix} \begin{bmatrix}x & 1\\ 1 & 0\end{bmatrix} =\begin{bmatrix} (x^{3}+2x)x+(x^{2}+1)\cdot1 & (x^{3}+2x)\cdot1+(x^{2}+1)\cdot0\\[4pt] (x^{2}+1)x+x\cdot1 & (x^{2}+1)\cdot1+x\cdot0 \end{bmatrix} =\begin{bmatrix} x^{4}+3x^{2}+1 & x^{3}+2x\\ x^{3}+2x & x^{2}+1 \end{bmatrix}.$$$

Thus the $$a_{11}$$ entry of $$A^4$$ is $$x^{4}+3x^{2}+1$$. We are told that $$a_{11}=109$$, so

$$$x^{4}+3x^{2}+1=109 \;\Longrightarrow\; x^{4}+3x^{2}-108=0.$$$

Letting $$y=x^{2}\;(y\ge 0)$$ converts this quartic into the quadratic

$$y^{2}+3y-108=0.$$

Using the quadratic-formula $$y=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$$ with $$a=1,\;b=3,\;c=-108$$, we find

$$$y=\dfrac{-3\pm\sqrt{3^{2}-4(1)(-108)}}{2}=\dfrac{-3\pm\sqrt{9+432}}{2}=\dfrac{-3\pm21}{2}.$$$

This gives $$y=\dfrac{18}{2}=9$$ or $$y=\dfrac{-24}{2}=-12$$. Because $$y=x^{2}\ge0$$, we discard the negative root and keep $$x^{2}=9$$, i.e. $$x=\pm3$$.

The $$a_{22}$$ entry of $$A^4$$ is, from our earlier result, $$x^{2}+1$$. Substituting $$x^{2}=9$$ yields

$$a_{22}=9+1=10.$$

So, the answer is $$10$$.