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Question Number 206600 by naka3546 last updated on 20/Apr/24

Answered by aleks041103 last updated on 22/Apr/24

$$\left[\begin{array}{ccc}-1& 3& 3\\ 1& 0& 0\\ 1& -3& -2\end{array}\right]\stackrel{3rdrow+1strow}{\to }\left[\begin{array}{ccc}-1& 3& 3\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$$$$\Rightarrow \left[\begin{array}{ccc}-1& 3& 3\\ 1& 0& 0\\ 1& -3& -2\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -1& 0& 1\end{array}\right]\left[\begin{array}{ccc}-1& 3& 3\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$$$$\left[\begin{array}{ccc}-1& 3& 3\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]\stackrel{1strow+2ndrow}{\to }\left[\begin{array}{ccc}0& 3& 3\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$$$$\Rightarrow \left[\begin{array}{ccc}-1& 3& 3\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& -1& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& 3& 3\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$$$$\left[\begin{array}{ccc}0& 3& 3\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]\stackrel{1strow-3\left(3rdrow\right)}{\to }\left[\begin{array}{ccc}0& 3& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$$$$\Rightarrow \left[\begin{array}{ccc}0& 3& 3\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 3\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& 3& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$$$$\left[\begin{array}{ccc}0& 3& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}3& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$$$$$$$$\Rightarrow \left[\begin{array}{ccc}-1& 3& 3\\ 1& 0& 0\\ 1& -3& -2\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ -1& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& -1& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 3\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}3& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right]$$

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