Question:
If $A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]$ and $A^{2}$ is the identity matrix, then $x=$ ____________
Solution:
The given matrix is $A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]$.
It is given that, $A^{2}=I$
$\therefore\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\Rightarrow\left[\begin{array}{cc}x^{2}+1 & x \\ x & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\Rightarrow x=0$
Thus, the value of $x$ is 0 . If $A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]$ and $A^{2}$ is the identity matrix, then $x=\underline{0}$