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Matrics — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsMatrics2 Marks
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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