We are asked to solve the following linear system
$$x_1-3x_2+x_3=1$$
$$2x_1-x_2-2x_3=2$$
$$x_1+2x_2-3x_3=-1$$
by using gauss-jordan elimination method. The augmented matrix of the linear system is $$\left(\begin{array}{ccc|c}1 & -3 & 1 & 1 \\2 & -1 & -2 & 2 \\1 & 2 & -3 & -1\end{array}\right).$$ By a series of elementary row operations, we have $$\left(\begin{array}{ccc|c}1 & -3 & 1 & 1 \\0 & 5 & -4 & 0 \\0 & 0 & 0 & -2\end{array}\right).$$ My question is, although the question asked us to solve the linear system using gauss-jordan elimination method, can we stop immediately and conclude that the linear system is inconsistent without further apply any elementary row operation to the matrix $$\left(\begin{array}{ccc|c}1 & -3 & 1 & 1 \\0 & 5 & -4 & 0 \\0 & 0 & 0 & -2\end{array}\right)$$ until the matrix $$\left(\begin{array}{ccc|c}1 & -3 & 1 & 1 \\0 & 5 & -4 & 0 \\0 & 0 & 0 & -2\end{array}\right)$$ is transformed into reduced-row echelon form?
Best Answer
You can conclude that the system is inconsistent, because the last row of your matrix implies that $0x_1+0x_2+0x_3=-2$, which cannot be satisfied.