Solve the homogeneous system. $$\left( \begin{array} { r }
2 x - y + 2 z = 0 \
x + 6 y + z = 0 \
x - 7 y + z = 0
\end{array} \right)$$
A) (0, 0, 0)
B) dependent
C) inconsistent

Question

Solve the homogeneous system. $$\left( \begin{array} { r } 
2 x - y + 2 z = 0 \
x + 6 y + z = 0 \
x - 7 y + z = 0
\end{array} \right)$$
A) (0, 0, 0)
B) dependent
C) inconsistent

Quizplus · Accepted Answer

We can use row reduction to find the solution:

$$\left( \begin{array} { r } 
2 & -1 & 2 \
1 & 6 & 1 \
1 & -7 & 1
\end{array} \right) \to \left( \begin{array} { r } 
1 & 6 & 1 \
2 & -1 & 2 \
1 & -7 & 1
\end{array} \right) \to \left( \begin{array} { r } 
1 & 6 & 1 \
0 & -13 & 0 \
0 & -13 & 0
\end{array} \right) \to \left( \begin{array} { r } 
1 & 6 & 1 \
0 & 1 & 0 \
0 & 0 & 0
\end{array} \right)$$

The last row represents the equation $0z=0$, which is always true. However, the second row gives us $y=0$, and the first row gives us $x=-6z$. Therefore, the set of solutions is of the form $(x,0,z)=(-6z, 0, z)=z(-6,0,1)$, which is a dependent set. Thus, the correct answer is B.

Solve the Homogeneous System (2x−y+2z=0x+6y+z=0x−7y+z=0)\left( \begin{array} { r } 2 x - y + 2 z = 0 \\x + 6 y + z = 0 \\x - 7 y + z = 0\end{array} \right)​2x−y+2z=0x+6y+z=0x−7y+z=0​​

Solve the Homogeneous System $\left( \begin{array} { r } 2 x - y + 2 z = 0 \\x + 6 y + z = 0 \\x - 7 y + z = 0\end{array} \right)$