Question

Find the inverse of the matrix, $A=\left[\begin{array}{ccc}1& 2& -2\\ -1& 3& 0\\ 0& -2& 1\end{array}\right]$ using elementary row transformations.

Answer 1

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

$|A|=\left|\begin{array}{ccc}1& 2& -2\\ -1& 3& 0\\ 0& -2& 1\end{array}\right|$

$=1\left(3\right)-2(-1)-2\left(2\right)$

$=3+2-4$

$=1$

$\therefore |A|\ne 0\Rightarrow A^{-1}$ exists.

We know $A{A}^{-1}=I$

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

$R_2\to R_2+R_1$

$\Rightarrow \left[\begin{array}{ccc}1& 2& -2\\ 0& 5& -2\\ 0& -2& 1\end{array}\right]A^{-1}=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 0& 0& 1\end{array}\right]$

$R_2\to R_2+2R_3$

$\Rightarrow \left[\begin{array}{ccc}1& 2& -2\\ 0& 1& 0\\ 0& -2& 1\end{array}\right]A^{-1}=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 2\\ 0& 0& 1\end{array}\right]$

$R_1\to R_1-2R_2,R_3\to R_3+2R_2$

$\Rightarrow \left[\begin{array}{ccc}1& 0& -2\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]A^{-1}=\left[\begin{array}{ccc}-1& -2& -4\\ 1& 1& 2\\ 2& 2& 5\end{array}\right]$

$R_1\to R_1+2R_3$

$\Rightarrow \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]A^{-1}=\left[\begin{array}{ccc}3& 2& 6\\ 1& 1& 2\\ 2& 2& 5\end{array}\right]$

$\therefore A^{-1}=\left[\begin{array}{ccc}3& 2& 6\\ 1& 1& 2\\ 2& 2& 5\end{array}\right]$