Let A be a $3 imes 3$ matrix such that $ \mathrm{A}\left[\begin{array}{lll} 1 & 2 & 3 \ 0 & 2 & 3 \ 0 & 1 & 1 \end{array} ight]=\left[\begin{array}{lll} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 \end{array} ight] $ Then $\mathrm{A}^{-1}$ is:

Question

Let A be a $3 	imes 3$ matrix such that $ \mathrm{A}\left[\begin{array}{lll} 1 & 2 & 3 \ 0 & 2 & 3 \ 0 & 1 & 1 \end{array}ight]=\left[\begin{array}{lll} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 \end{array}ight] $ Then $\mathrm{A}^{-1}$ is:

Aicharya · Accepted Answer

$ 	ext { Given } A\left[\begin{array}{lll} 1 & 2 & 3 \ 0 & 2 & 3 \ 0 & 1 & 1 \end{array}ight]=\left[\begin{array}{lll} 0 & 0 & 1 \ 1 & 0 & 0 \ 0 & 1 & 0 \end{array}ight] $ Applying $\mathrm{C}_1 \leftrightarrow \mathrm{C}_3$ $ \mathrm{A}\left[\begin{array}{lll} 3 & 2 & 1 \ 3 & 2 & 0 \ 1 & 1 & 0 \end{array}ight]=\left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 \end{array}ight] $ Again Applying $\mathrm{C}_2 \leftrightarrow \mathrm{C}_3$ $ 	ext { A }\left[\begin{array}{lll} 3 & 1 & 2 \ 3 & 0 & 2 \ 1 & 0 & 1 \end{array}ight]=\left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}ight] $ pre-multiplying both sides by $\mathrm{A}^{-1}$ $ \begin{aligned} &\mathrm{A}^{-1} \mathrm{~A}\left[\begin{array}{lll} 3 & 1 & 2 \ 3 & 0 & 2 \ 1 & 0 & 1 \end{array}ight]=\mathrm{A}^{-1}\left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}ight] \ &\mathrm{I}\left[\begin{array}{lll} 3 & 1 & 2 \ 3 & 0 & 2 \ 1 & 0 & 1 \end{array}ight]=\mathrm{A}^{-1} \mathrm{I}=\mathrm{A}^{-1} \ &\left(\because \mathrm{A}^{-1} \mathrm{~A}=\mathrm{I} 	ext { and } \mathrm{I}=	ext { Identity matrix }ight) \end{aligned} $ $ 	ext { Hence, } A^{-1}=\left[\begin{array}{lll} 3 & 1 & 2 \ 3 & 0 & 2 \ 1 & 0 & 1 \end{array}ight] $

Let A be a $3 \times 3$ matrix such that $ \mathrm{A}\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 &…

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