Question

If possible, using elementary row transformations, find the inverse of the following matrices.
$\left[\begin{array}{ccc}2& 3& -3\\ -1& -2& 2\\ 1& 1& -1\end{array}\right]$

Answer 1

For getting the inverse of the given matrix A by row elementary operations, we may write the given matrix as A = IA

(ii) $\therefore \left[\begin{array}{ccc}2& 3& -3\\ -1& -2& 2\\ 1& 1& -1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]A$

$\Rightarrow \left[\begin{array}{ccc}0& 1& -1\\ 0& -1& 1\\ 1& 1& -1\end{array}\right]=\left[\begin{array}{ccc}1& 0& -2\\ 0& 1& 1\\ 0& 0& 1\end{array}\right]A$ $\left[\begin{array}{c}\because R_2\to R_2+R_3\\ and{R}_1\to R_1-2R_3\end{array}\right]$

$\left[\begin{array}{ccc}0& 1& -1\\ 0& 0& 0\\ 1& 1& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& -2\\ 2& 1& -2\\ 0& 0& 1\end{array}\right]A$ $[\therefore R_2\to R_2+R_1]$

Since, second row of the matrix A on LHS is containing all zeroes, so we can say that inverse of matrix A does not exist.