Question

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

Answer 1

$A=\begin{bmatrix}x&1\\ 1&0\end{bmatrix}$
$A^{2} = \begin{bmatrix}x&1\\ 1&0\end{bmatrix}\begin{bmatrix}x&1\\ 1&0\end{bmatrix} = \begin{bmatrix}x^{2} +1&x\\ x&1\end{bmatrix}$
$A^{4} = \begin{bmatrix}x^{2} +1&x\\ x&1\end{bmatrix}\begin{bmatrix}x^{2} +1&x\\ x&1\end{bmatrix}$
$= \begin{bmatrix}\left(x^{2}+1\right)^{2} +x^{2}&x\left(x^{2} +1\right) +x\\ x\left(x^{2} +1\right)+x&x^{2} +1\end{bmatrix}$
$a_{11}=\left(x^{2}+1\right)^{2}+x^{2}=109$
$\Rightarrow x=\pm 3$
$a_{22}=x^{2}+1=10$