Q. 52

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.

$\{\begin{cases} 2 x - 3 y - z = 0 \\ - x + 2 y + z = 5 \\ 3 x - 4 y - z = 1 \end{cases}$

Verified

Answer

The system of equation is inconsistent.

1Step 1. Given information.

The given system of equation are

$\{\begin{cases} 2 x - 3 y - z = 0 \\ - x + 2 y + z = 5 \\ 3 x - 4 y - z = 1 \end{cases}$

2Step 2. Calculation.

The augmented matrix of the system is:

$[\begin{matrix} \begin{matrix} 2 & - 3 & - 1 \\ - 1 & 2 & 1 \\ 3 & - 4 & - 1 \end{matrix} & \begin{matrix} 0 \\ 5 \\ 1 \end{matrix} \end{matrix}]$

Perform the row operations $R_{1} = r_{1} + 2 r_{2}; R_{3} = r_{3} + 3 r_{2}$

$[\begin{matrix} \begin{matrix} 0 & 1 & 1 \\ - 1 & 2 & 1 \\ 0 & 2 & 2 \end{matrix} & \begin{matrix} 10 \\ 5 \\ 16 \end{matrix} \end{matrix}]$

Perform the row operations $R_{3} = r_{3} - 2 r_{1}$ :

$[\begin{matrix} \begin{matrix} 0 & 1 & 1 \\ - 1 & 2 & 1 \\ 0 & 0 & 0 \end{matrix} & \begin{matrix} 10 \\ 5 \\ - 4 \end{matrix} \end{matrix}]$

Use the obtained matrix to write the system of equations.

$y + z = 10 - x + y + z = 5 0 = - 4$

This system is impossible to solve. Therefore, this system of equation is inconsistent.

3Step 3. Conclusion.

The system of equation is inconsistent.