91Ó°ÊÓ

The given system of equation is

$\left\{\begin{array}{l}x+2y-z=3\\ 2x-y+2z=6\\ x-3y+3z=4\end{array}\right.$

Row operation :

1. Interchange any two rows.
2. Replace a row by a nonzero multiple of that row.
3. Replace a row by the sum of that row and a constant nonzero multiple of some other row.

The augmented matrix of the system is:

$\left[\begin{array}{cc}\begin{array}{ccc}1& 2& -1\\ 2& -1& 2\\ 1& -3& 3\end{array}& \begin{array}{c}3\\ 6\\ 4\end{array}\end{array}\right]$

Perform the row operations $R_2=r_2-2r_1$:

$\left[\begin{array}{cc}\begin{array}{ccc}1& 2& -1\\ 0& -5& 4\\ 1& -3& 3\end{array}& \begin{array}{c}3\\ 0\\ 4\end{array}\end{array}\right]$

Perform the row operations $R_3=r_3-r_1$:

role="math" localid="1648038161972" $\left[\begin{array}{cc}\begin{array}{ccc}1& 2& -1\\ 0& -5& 4\\ 0& -5& 4\end{array}& \begin{array}{c}3\\ 0\\ 1\end{array}\end{array}\right]$

Perform the row operations $R_2=-\frac{r_2}{5}$:

role="math" localid="1648038361196" $\left[\begin{array}{cc}\begin{array}{ccc}1& 2& -1\\ 0& 1& -\frac{4}{5}\\ 0& -5& 4\end{array}& \begin{array}{c}3\\ 0\\ 1\end{array}\end{array}\right]$

Perform the row operations $R_1=r_1-2r_2$:

role="math" localid="1648038368722" $\left[\begin{array}{cc}\begin{array}{ccc}1& 0& \frac{3}{5}\\ 0& 1& -\frac{4}{5}\\ 0& -5& 4\end{array}& \begin{array}{c}3\\ 0\\ 1\end{array}\end{array}\right]$

Perform the row operations $R_3=r_3+5r_2$:

$\left[\begin{array}{cc}\begin{array}{ccc}1& 0& \frac{3}{5}\\ 0& 1& -\frac{4}{5}\\ 0& 0& 0\end{array}& \begin{array}{c}3\\ 0\\ 1\end{array}\end{array}\right]$

This matrix is in row echelon form.

Use the obtained matrix to write the system of equations.

role="math" localid="1648038564733" $x+\frac{3}{5}z=3...\left(1\right)$

$y-\frac{4}{5}z=0...\left(2\right)$

From the above obtained matrix, the bottom row is equivalent to the equation

$0x+0y+0z=1$

which has no solution.

Therefore, the original system is inconsistent.

The given system is inconsistent.