Question

If $A=\left[\begin{matrix}x&1\\ 1&0\end{matrix}\right]$ and $A=A^{-1}$, then $x = .......$.

• A. $0$
• B. $4$
• C. $2$
• D. $1$

Answer 1

Answer: (A)

We have, $A=\begin{bmatrix}x&1\\ 1&0\end{bmatrix}$
$\therefore \left|A\right| =\left(x \times0-1\right)= -1$
adj $A= \begin{bmatrix}0&-1\\ -1&x\end{bmatrix}$
$A^{-1} = \frac{adjA}{\left|A\right|} = \left(-1\right) \begin{bmatrix}0&-1\\ -1&x\end{bmatrix} =\begin{bmatrix}0&1\\ 1&-x\end{bmatrix}$
Since, It is given, $A = A^{-1}$
$\Rightarrow \begin{bmatrix}x&1\\ 1&0\end{bmatrix}=\begin{bmatrix}0&1\\ 1&-x\end{bmatrix}$
$\Rightarrow x=0$ (By equality of matrices)