Let A = $\left[ {\matrix{ x & 1 \cr 1 & 0 \cr } } \right]$, x $\in$ R and A⁴ = [a_ij].
If a₁₁ = 109, then a₂₂ is equal to _______ .

$${A^2} = \left[ {\matrix{ x &amp; 1 \cr 1 &amp; 0 \cr } } \right]\left[ {\matrix{ x &amp; 1 \cr 1 &amp; 0 \cr } } \right] = \left[ {\matrix{ {{x^2} + 1} &amp; x \cr x &amp; 1 \cr } } \right]$$<br><br>$${A^4} = \left[ {\matrix{ {{x^2} + 1} &amp; x \cr x &amp; 1 \cr } } \right]\left[ {\matrix{ {{x^2} + 1} &amp; x \cr x &amp; 1 \cr } } \right]$$<br><br>$$ = \left[ {\matrix{ {{{({x^2} + 1)}^2} + {x^2}} &amp; {x({x^2} + 1) + x} \cr {x({x^2} + 1) + x} &amp; {{x^2} + 1} \cr } } \right]$$<br><br>Given ${({x^2} + 1)^2} + {x^2} = 109$<br><br>Let ${x^2} + 1$ = t<br><br>${t^2} + t - 1 = 109$<br><br>$\Rightarrow$ (t $-$ 10) (t + 11) = 0<br><br>$\therefore$ t = 10 = x² + 1 = a₂₂