If $A^{-1}=\left[\begin{array}{lll}3 & 2 & 6 \ 1 & 1 & 2 \ 2 & 5 & 5\end{array} ight]$, then $A=$

Question

If $A^{-1}=\left[\begin{array}{lll}3 & 2 & 6 \ 1 & 1 & 2 \ 2 & 5 & 5\end{array}ight]$, then $A=$

Aicharya · Accepted Answer

Let $\mathrm{A}^{-1} \mathrm{~A}=\mathrm{I}$ $\begin{aligned} & {\left[\begin{array}{lll} 3 & 2 & 6 \ 1 & 1 & 2 \ 2 & 5 & 5 \end{array}ight] A=\left[\begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}ight]} \ & R_2 ightarrow 3 R_2-R_1 	ext { and } R_3 ightarrow 3 R_3-2 R_1 \ & {\left[\begin{array}{ccc} 3 & 2 & 6 \ 0 & 1 & 0 \ 0 & 11 & 3 \end{array}ight] A=\left[\begin{array}{ccc} 1 & 0 & 0 \ -1 & 3 & 0 \ -2 & 0 & 3 \end{array}ight]} \ & R_1 ightarrow R_1-2 R_2 	ext { and } R_3 ightarrow R_3-11 R_2 \end{aligned}$ $\begin{aligned} & {\left[\begin{array}{lll} 3 & 0 & 6 \ 0 & 1 & 0 \ 0 & 0 & 3 \end{array}ight] A=\left[\begin{array}{ccc} 3 & -6 & 0 \ -1 & 3 & 0 \ 9 & -33 & 3 \end{array}ight]} \ & \mathrm{R}_1 ightarrow \mathrm{R}_1-2 \mathrm{R}_3 \ & {\left[\begin{array}{lll} 3 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 3 \end{array}ight] \mathrm{A}=\left[\begin{array}{ccc} -15 & 60 & -6 \ -1 & 3 & 0 \ 9 & -33 & 3 \end{array}ight]} \ & \mathrm{R}_1 ightarrow \frac{1}{3} \mathrm{R}_1 	ext { and } \mathrm{R}_3 ightarrow \frac{1}{3} \mathrm{R}_3 \ & {\left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}ight] \mathrm{A}=\left[\begin{array}{ccc} -5 & 20 & -2 \ -1 & 3 & 0 \ 3 & -11 & 1 \end{array}ight]} \end{aligned}$

If $A^{-1}=\left[\begin{array}{lll}3 & 2 & 6 \\ 1 & 1 & 2 \\ 2 & 5 & 5\end{array}\right]$, then $A=$

Solution