Question

Find the inverse of each of the following matrices by using elementary row transformations:

$\left[\begin{array}{ccc}2& -1& 3\\ 1& 2& 4\\ 3& 1& 1\end{array}\right]$

Answer 1

$$\begin{gathered} A=\left[\begin{array}{ccc}2& -1& 3\\ 1& 2& 4\\ 3& 1& 1\end{array}\right] \\ \mathrm{We}\mathrm{know} \\ A=IA \\ \Rightarrow \left[\begin{array}{ccc}2& -1& 3\\ 1& 2& 4\\ 3& 1& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]A \\ \Rightarrow \left[\begin{array}{ccc}1& -\frac{1}{2}& \frac{3}{2}\\ 1& 2& 4\\ 3& 1& 1\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{2}& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]A\left[\mathrm{Applying}{R}_1\to \frac{1}{2}{R}_1\right] \\ \Rightarrow \left[\begin{array}{ccc}1& -\frac{1}{2}& \frac{3}{2}\\ 0& \frac{5}{2}& \frac{5}{2}\\ 0& \frac{5}{2}& \frac{-7}{2}\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{2}& 0& 0\\ -\frac{1}{2}& 1& 0\\ -\frac{3}{2}& 0& 1\end{array}\right]A\left[\mathrm{Applying}{R}_2\to R_2-R_1\mathrm{and}{R}_3\to R_3-3R_1\right] \\ \Rightarrow \left[\begin{array}{ccc}1& -\frac{1}{2}& \frac{3}{2}\\ 0& 1& 1\\ 0& \frac{5}{2}& \frac{-7}{2}\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{2}& 0& 0\\ -\frac{1}{5}& \frac{2}{5}& 0\\ -\frac{3}{2}& 0& 1\end{array}\right]A\left[\mathrm{Applying}{R}_2\to \frac{2}{5}{R}_2\right] \\ \Rightarrow \left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 1\\ 0& 0& -6\end{array}\right]=\left[\begin{array}{ccc}\frac{2}{5}& \frac{1}{5}& 0\\ -\frac{1}{5}& \frac{2}{5}& 0\\ -1& -1& 1\end{array}\right]A\left[\mathrm{Applying}{R}_1\to R_1+\frac{1}{2}{R}_2\mathrm{and}{R}_3\to R_3-\frac{5}{2}{R}_2\right] \\ \Rightarrow \left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}\frac{2}{5}& \frac{1}{5}& 0\\ -\frac{1}{5}& \frac{2}{5}& 0\\ \frac{1}{6}& \frac{1}{6}& -\frac{1}{6}\end{array}\right]A\left[\mathrm{Applying}{R}_3\to -\frac{1}{6}{R}_2\right] \\ \Rightarrow \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{15}& \frac{-2}{15}& \frac{1}{3}\\ -\frac{11}{30}& \frac{7}{30}& \frac{1}{6}\\ \frac{1}{6}& \frac{1}{6}& -\frac{1}{6}\end{array}\right]A\left[\mathrm{Applying}{R}_2\to R_2-R_3\mathrm{and}{R}_1\to R_1-2R_3\right] \\ \Rightarrow A^{-1}=-\frac{1}{30}\left[\begin{array}{ccc}-2& 4& -10\\ 11& -7& -5\\ -5& -5& 5\end{array}\right] \end{gathered}$$