JEE Main 2014 — Mathematics Algebra

$$\text { Given } A\left[\begin{array}{lll} 1 & 2 & 3 \\ 0 & 2 & 3 \\ 0 & 1 & 1 \end{array}\right]=\left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right]$$ Given A​100​221​331​​=​010​001​100​​ Applying $\mathrm{C}_1 \leftrightarrow \mathrm{C}_3$C1​↔C3​ $$\mathrm{A}\left[\begin{array}{lll} 3 & 2 & 1 \\ 3 & 2 & 0 \\ 1 & 1 & 0 \end{array}\right]=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]$$A​331​221​100​​=​100​001​010​​ Again Applying $\mathrm{C}_2 \leftrightarrow \mathrm{C}_3$C2​↔C3​ $$\text { A }\left[\begin{array}{lll} 3 & 1 & 2 \\ 3 & 0 & 2 \\ 1 & 0 & 1 \end{array}\right]=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$$ A ​331​100​221​​=​100​010​001​​ pre-multiplying both sides by $\mathrm{A}^{-1}$A−1 $$\begin{aligned} &\mathrm{A}^{-1} \mathrm{~A}\left[\begin{array}{lll} 3 & 1 & 2 \\ 3 & 0 & 2 \\ 1 & 0 & 1 \end{array}\right]=\mathrm{A}^{-1}\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \\ &\mathrm{I}\left[\begin{array}{lll} 3 & 1 & 2 \\ 3 & 0 & 2 \\ 1 & 0 & 1 \end{array}\right]=\mathrm{A}^{-1} \mathrm{I}=\mathrm{A}^{-1} \\ &\left(\because \mathrm{A}^{-1} \mathrm{~A}=\mathrm{I} \text { and } \mathrm{I}=\text { Identity matrix }\right) \end{aligned}$$​A−1 A​331​100​221​​=A−1​100​010​001​​I​331​100​221​​=A−1I=A−1(∵A−1 A=I and I= Identity matrix )​ $$\text { Hence, } A^{-1}=\left[\begin{array}{lll} 3 & 1 & 2 \\ 3 & 0 & 2 \\ 1 & 0 & 1 \end{array}\right]$$ Hence, A−1=​331​100​221​​