Question

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

$\left[\begin{array}{ccc}3& 0& -1\\ 2& 3& 0\\ 0& 4& 1\end{array}\right]$

Answer 1

$$\begin{gathered} A=\left[\begin{array}{ccc}3& 0& -1\\ 2& 3& 0\\ 0& 4& 1\end{array}\right] \\ \mathrm{We}\mathrm{know} \\ A=IA \\ \Rightarrow \left[\begin{array}{ccc}3& 0& -1\\ 2& 3& 0\\ 0& 4& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]A \\ \Rightarrow \left[\begin{array}{ccc}1& 0& -\frac{1}{3}\\ 2& 3& 0\\ 0& 4& 1\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{3}& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]A\left[\mathrm{Applying}{R}_1\to \frac{1}{3}{R}_2\right] \\ \Rightarrow \left[\begin{array}{ccc}1& 0& -\frac{1}{3}\\ 0& 3& \frac{2}{3}\\ 0& 4& 1\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{3}& 0& 0\\ -\frac{2}{3}& 1& 0\\ 0& 0& 1\end{array}\right]A\left[\mathrm{Applying}{R}_2\to R_2-2R_1\right] \\ \Rightarrow \left[\begin{array}{ccc}1& 0& -\frac{1}{3}\\ 0& 1& \frac{2}{9}\\ 0& 4& 1\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{3}& 0& 0\\ -\frac{2}{9}& \frac{1}{3}& 0\\ 0& 0& 1\end{array}\right]A\left[\mathrm{Applying}{R}_2\to \frac{1}{3}{R}_2\right] \\ \Rightarrow \left[\begin{array}{ccc}1& 0& -\frac{1}{3}\\ 0& 1& \frac{2}{9}\\ 0& 0& \frac{1}{9}\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{3}& 0& 0\\ -\frac{2}{9}& \frac{1}{3}& 0\\ \frac{8}{9}& \frac{-4}{3}& 1\end{array}\right]A\left[\mathrm{Applying}{R}_3\to R_3-4R_2\right] \\ \Rightarrow \left[\begin{array}{ccc}1& 0& -\frac{1}{3}\\ 0& 1& \frac{2}{9}\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{3}& 0& 0\\ -\frac{2}{9}& \frac{1}{3}& 0\\ 8& -12& 9\end{array}\right]A\left[\mathrm{Applying}{R}_3\to 9R_3\right] \\ \Rightarrow \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}3& -4& 3\\ -2& 3& -2\\ 8& -12& 9\end{array}\right]A\left[\mathrm{Applying}{R}_2\to R_2-\frac{2}{9}{R}_3\mathrm{and}{R}_1\to R_1+\frac{1}{3}{R}_3\right] \\ \therefore A^{-1}=\left[\begin{array}{ccc}3& -4& 3\\ -2& 3& -2\\ 8& -12& 9\end{array}\right] \end{gathered}$$