If $A=\left[\begin{array}{ll}2 & -2 \\ 2 & -3\end{array}\right], B=\left[\begin{array}{cc}0 & -1 \\ 1 &…

If $A=\left[\begin{array}{ll}2 & -2 \\ 2 & -3\end{array}\right], B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$, then $\left(B^{-1} A^{-1}\right)^{-1}=$ ?
  1. $A=\left[\begin{array}{ll}-2 & -2 \\ -3 & -2\end{array}\right]$
  2. $A=\left[\begin{array}{cc}2 & 2 \\ -2 & -3\end{array}\right]$
  3. $A=\left[\begin{array}{cc}3 & -2 \\ 2 & 2\end{array}\right]$
  4. $A=\left[\begin{array}{cc}1 & -1 \\ -2 & 3\end{array}\right]$

Solution

$\begin{aligned} & \left(B^{-1} A^{-1}\right)^{-1}=\left(A^{-1}\right)^{-1}\left(B^{-1}\right)^{-1}=A B \\ & \therefore\left(B^{-1} A^{-1}\right)^{-1}=\left[\begin{array}{ll}2 & -2 \\ 2 & -3\end{array}\right]\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]=\left[\begin{array}{ll}-2 & -2 \\ -3 & -2\end{array}\right]\end{aligned}$

Asked in: MHT CET 2021 (23 Sep Shift 1)

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