question_answer

Using elementary transformations, find the inverse of the matrix \[\left[ \begin{matrix}    3 & 10  \\    2 & 7  \\ \end{matrix} \right],\] if it exists. 

Answer:

Let \[A=\left[ \begin{matrix}    3 & 10  \\    2 & 7  \\ \end{matrix} \right]\] For applying elementary row operations, consider the matrix equation             \[\left[ \begin{matrix}    3 & 10  \\    2 & 7  \\ \end{matrix} \right]=\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]A\]                             Applying \[{{R}_{1}}\to {{R}_{1}}-2{{R}_{2}},\] we get                \[\left[ \begin{matrix}    1 & 3  \\    2 & 7  \\ \end{matrix} \right]=\left[ \begin{matrix}    1 & -\,1  \\    0 & 1  \\ \end{matrix} \right]A\]            Applying \[{{R}_{2}}\to {{R}_{2}}-2{{R}_{1}},\] we get                                \[\left[ \begin{matrix}    1 & 3  \\    0 & 1  \\ \end{matrix} \right]=\left[ \begin{matrix}    1 & -\,1  \\    -\,2 & 3  \\ \end{matrix} \right]A\] Applying \[{{R}_{1}}\to {{R}_{1}}-3{{R}_{2}},\] we get                    \[\left[ \begin{matrix}    1 & 0  \\    0 & 1  \\ \end{matrix} \right]=\left[ \begin{matrix}    7 & -\,10  \\    -\,2 & 3  \\ \end{matrix} \right]A\] \[\Rightarrow \] \[{{A}^{-1}}=\left[ \begin{matrix}    7 & -\,10  \\    -\,2 & 3  \\ \end{matrix} \right]\]