Q76 / 155 · WBJEE 2024 · Mathematics
If $\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] \cdot A \cdot\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$, then $A=$
- Correct answerA$\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]$
- B$\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$
- C$\left[\begin{array}{ll}1 & 0 \\ 1 & 1\end{array}\right]$
- D$\left[\begin{array}{ll}0 & 1 \\ 1 & 1\end{array}\right]$
Solution
\(\begin{aligned} & {\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right] A\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]} \\ & A=\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right]^{-1}\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}-3 & 2 \\ 5 & -3\end{array}\right]^{-1} \\ & \Rightarrow A=\left[\begin{array}{ccc}2 & -1 \\ -3 & 2\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}3 & 2 \\ 5 & 3\end{array}\right] \\ & =\left[\begin{array}{cc}2 & -1 \\ -3 & 2\end{array}\right]\left[\begin{array}{ll}3 & 2 \\ 5 & 3\end{array}\right]=\left[\begin{array}{ll}1 & 1 \\ 1 & 0\end{array}\right]\end{aligned}\)
Hence, the correct answer is option: 'A'.