Question

Is there a sequence of elementary row operations that transforms

$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$into $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$?

Short answer

No, there is not any sequence of operation that transform$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ into $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$.

Step-by-step solution

Row operations for changing the matrix

We have given a matrix for transforming the matrix we will follow the following operations.

$$\begin{gathered} \left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right] \\ {R}_2\to R_2-4R_1 \\ {R}_3\to R_3-7R_1 \\ ~\left[\begin{array}{ccc}1& 2& 3\\ 0& -3& -6\\ 0& -6& -12\end{array}\right] \\ {R}_2\to \frac{1}{-3}{R}_2 \\ {R}_3\to \frac{1}{-6}{R}_3 \\ ~\left[\begin{array}{ccc}1& 2& 3\\ 0& 1& 2\\ 0& 1& 2\end{array}\right] \end{gathered}$$

Solving matrix

Now proceeding further with more sequence of operation

$$\begin{gathered} R_1\to R_1-2R_2 \\ {R}_3\to R_3-R_2 \end{gathered}$$

Using these above operations we get,

$~\left[\begin{array}{ccc}1& 0& -1\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]\ne \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$

Hence there is not any sequence of operation that transform$\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]$ into $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$.