A = $\left(\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right)$
$\mathit{A}{\mathit{A}}^{-1}$ = I
$\left(\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right)$ ${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$
${\mathit{R}}_1$ ↔ ${\mathit{R}}_3$
$\left(\begin{array}{ccc}1& 1& -2\\ 3& 2& -4\\ 2& -& 3& 5\end{array}\right)$ ${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$
${\mathit{R}}_2$ → ${\mathit{R}}_2-3{\mathit{R}}_1$ . ${\mathit{R}}_3$ → ${\mathit{R}}_3-2{\mathit{R}}_1$
$\left(\begin{array}{ccc}1& 1& -2\\ 0& -1& 2\\ 0& -5& 9\end{array}\right)$ ${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}0& 0& 1\\ 0& 1& -3\\ 1& 0& -2\end{array}\right)$
${\mathit{R}}_2$ → - ${\mathit{R}}_2$
$\left(\begin{array}{ccc}1& 1& -2\\ 0& -1& 2\\ 0& -5& 9\end{array}\right)$ ${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}0& 0& 1\\ 0& -1& 3\\ 1& 0& -2\end{array}\right)$
${\mathit{R}}_1$ → ${\mathit{R}}_1-{\mathit{R}}_2$ , ${\mathit{R}}_3$ → ${\mathit{R}}_3+5{\mathit{R}}_2$
$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& -2\\ 0& 0& -1\end{array}\right)$ ${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}0& 1& -2\\ 0& -1& 3\\ 1& -5& 13\end{array}\right)$
${\mathit{R}}_3$ → - ${\mathit{R}}_3$
$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& -2\\ 0& 0& -1\end{array}\right)$ ${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}0& 1& -2\\ 0& -1& 3\\ -1& 5& -13\end{array}\right)$
${\mathit{R}}_2$ → ${\mathit{R}}_2+2{\mathit{R}}_3$
$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$ ${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}0& 1& -2\\ -2& 9& -23\\ -1& 5& -13\end{array}\right)$
${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}0& 1& -2\\ -2& 9& -23\\ -1& 5& -13\end{array}\right)$
Consider,
AX = B where B = $\left[\begin{array}{c}11\\ -5\\ 3\end{array}\right]$ and X = $\left[\begin{array}{c}\mathit{x}\\ \mathit{y}\\ \mathit{z}\end{array}\right]$
⇒ ${\mathit{A}}^{-1}$ AX = ${\mathit{A}}^{-1}$ B
⇒ X = ${\mathit{A}}^{-1}$ B
⇒ X = $\left(\begin{array}{ccc}0& 1& -2\\ -2& 9& -23\\ -1& 5& -13\end{array}\right)$ $\left[\begin{array}{c}11\\ -5\\ 3\end{array}\right]$
⇒ $\left[\begin{array}{c}11\\ -5\\ 3\end{array}\right]$ = $\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$
⇒ x = 1 , y = 2 , z = 3
OR
Given A = $\left(\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right)$
Consider,
$\left|\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right|$ = 1 (- 25 + 28) - 2 (- 10 + 14) + 3 (- 8 + 10)
= 3 - 8 + 6
= 1 ≠ 0
${\mathit{A}}^{-1}$ exist.
A ${\mathit{A}}^{-1}$ = I
$\left(\begin{array}{ccc}1& 2& 3\\ 2& 5& 7\\ -2& -4& -5\end{array}\right)$ ${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$
${\mathit{R}}_2$ → ${\mathit{R}}_2-2{\mathit{R}}_1,{\mathit{R}}_3$ → ${\mathit{R}}_3+2{\mathit{R}}_1$
$\left(\begin{array}{ccc}1& 2& 3\\ 0& 1& 1\\ 0& 0& 1\end{array}\right)$ ${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}1& 0& 0\\ -2& 1& 0\\ 2& 0& 1\end{array}\right)$
${\mathit{R}}_1$ → ${\mathit{R}}_1-2{\mathit{R}}_2$
$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$ ${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}5& -2& 0\\ -2& 1& 0\\ 2& 0& 1\end{array}\right)$
${\mathit{R}}_1$ → ${\mathit{R}}_1-{\mathit{R}}_3,{\mathit{R}}_2$ → ${\mathit{R}}_2-{\mathit{R}}_3$
$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$ ${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}3& -2& -1\\ -4& 1& -1\\ 2& 0& -1\end{array}\right)$
${\mathit{A}}^{-1}$ = $\left(\begin{array}{ccc}3& -2& -1\\ -4& 1& -1\\ 2& 0& 1\end{array}\right)$