If A = [ ll x & 1 1 & 0 ] and A ^2= I, then A ^ -1 is equal to - Vidyadip

MCQ

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

✓
$\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$
B
$\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
C
$\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$
D
$\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

✓

Answer

Correct option: A.

$\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$

(A) $A ^2=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{cc}x^2+1 & x \\ x & 1\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$\Rightarrow x=0$
$\therefore A=\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$
$A ^2= I$
$\therefore \quad A^{-1} A \cdot A=A^{-1} I$
$\Rightarrow I \cdot A = A ^{-1} I$
$\Rightarrow A ^{-1}= A$

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