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Problem 14: Determine whether the following system of equations are consistent or not. If yes, find all possible solutions

Problem 14: Determine whether the following system of equations are consistent or not. If yes, find all possible solutions:

\begin{align*} x_1-2x_2-x_3+3x_4=0\\ -2x_1+4x_2+5x_3-5x_4=3\\ 3x_1-6x_2-6x_3+8x_4=2 \end{align*}

Solution:

Consider the augmented matrix of this system and apply row operations.

\[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrrr|r} 1 & -2 & -1& 3 & 0 \\ -2 & 4 & 5& -5 & 3 \\ 3 & -6 & -6& 8 & 2 \\ \end{array}\right] \xrightarrow{R2\rightarrow R_2+2R_1,R3\rightarrow R_3-3R_1} \left[\begin{array}{rrrr|r} 1 & -2 & -1& 3 & 0 \\ 0 & 0 & 3& 1 & 3 \\ 0 & 0 & -3& -1 & 2 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R3\rightarrow R_3+R_2} \left[\begin{array}{rrrr|r} 1 & -2 & -1& 3 & 0 \\ 0 & 0 & 3& 1 & 3 \\ 0 & 0 & 0& 0 & 5 \\ \end{array}\right] \xrightarrow{R_2\rightarrow \frac{1}{3}R_2} \left[\begin{array}{rrrr|r} 1 & -2 & -1& 3 & 0 \\ 0 & 0 & 1& \frac{1}{3} & 1 \\ 0 & 0 & 0& 0 & 5 \\ \end{array}\right]\] \[\xrightarrow{R1\rightarrow R_1+R_2} \left[\begin{array}{rrrr|r} 1 & -2 & 0& \frac{10}{3} & 1 \\ 0 & 0 & 1& \frac{1}{3} & 1 \\ 0 & 0 & 0& 0 & 5 \\ \end{array}\right]\]

rank of coefficient matrix = 2,rank of augmented matrix = 3. Since rank of coefficient matrix $\ne$ rank of augmented matrix, the system is inconsistent.

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