Question

Find inverse of the matrix $\left[\begin{array}{ccc}1& -1& 1\\ 0& 1& 1\\ 3& 2& -4\end{array}\right]$ by elementary transformation.

Answer 1

Inverse of $A\left[\begin{array}{ccc}1& -1& 1\\ 0& 1& 1\\ 3& 2& -4\end{array}\right]$ by elementary method.

Write idntity matrix at the side of given matrix.

$\left[\begin{array}{cccccc}1& -1& 1& 1& 0& 0\\ 0& 1& 1& 0& 1& 0\\ 3& 2& -4& 0& 0& 1\end{array}\right]\begin{array}{c}\to a\\ \to b\\ \to c\end{array}$

$\left[\begin{array}{cccccc}1& -1& 1& 1& 0& 0\\ 3& 0& 2& 0& -2& 1\\ 5& 0& -2& 2& 0& 1\end{array}\right]\begin{array}{c}a\to a\\ 2b+c=\to b\\ c+2a\to c\end{array}$

$\left[\begin{array}{cccccc}1& -1& 1& 1& 0& 0\\ 1& 0& 0& \frac{1}{4}& \frac{-1}{4}& \frac{1}{4}\\ 5& 0& -2& 2& 0& 1\end{array}\right]\begin{array}{c}a\to a\\ \frac{b+c}{8}=\to b\\ c\to c\end{array}$

$\left[\begin{array}{cccccc}7& -2& 0& 4& 0& 1\\ 1& 0& 0& \frac{1}{4}& \frac{-1}{4}& \frac{1}{4}\\ 0& 0& 1& \frac{-3}{8}& \frac{-5}{8}& \frac{1}{8}\end{array}\right]\begin{array}{c}2a+c\to a\\ b\to b\\ \frac{5b-c}{2}\to c\end{array}$

$\left[\begin{array}{cccccc}0& 1& 0& \frac{-9}{8}& \frac{-7}{8}& \frac{3}{8}\\ 1& 0& 0& \frac{1}{4}& \frac{-1}{4}& \frac{1}{4}\\ 0& 0& 1& \frac{-3}{8}& \frac{-5}{8}& \frac{1}{8}\end{array}\right]\begin{array}{c}\frac{7b-a}{2}\to a\\ b\to b\\ c\to c\end{array}$

$\left[\begin{array}{cccccc}1& 0& 0& \frac{1}{4}& \frac{-1}{4}& \frac{1}{4}\\ 0& 1& 0& \frac{-9}{8}& \frac{-7}{8}& \frac{3}{8}\\ 0& 0& 1& \frac{-3}{8}& \frac{-5}{8}& \frac{1}{8}\end{array}\right]\begin{array}{c}b\to a\\ a\to b\\ c\to c\end{array}$

$A^{-1}=\frac{1}{8}\left[\begin{array}{ccc}2& -2& 2\\ -9& -7& 3\\ -3& -5& 1\end{array}\right]$