Question

If $\left[\begin{array}{cc}x& 1\\ 1& 0\end{array}\right]$ and $A^2=I$, then $x=$

• A. $0$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

Answer: (A)

$0$

Given $A=\left[\begin{array}{cc}x& 1\\ 1& 0\end{array}\right]$

Also, $A^2=I$

$\Rightarrow \left[\begin{array}{cc}x& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}x& 1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}{x}^2+1& x\\ x& 1\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

For the above two matrices to be equal each of the first matrix must be equal to the corresponding element of the second matrix.

Therefore, $x^2+1=1...\left(i\right)$ and $x=\mathrm{0...}\left(ii\right)$.

From equation $\left(i\right)$ and equation $\left(ii\right)$ we get,

$x=0$

Hence, the correct option is $\left(A\right)$