Solve the homogeneous system. $$\left( \begin{array} { r }
x - 2 y + 5 z = 0 \
3 x + y - 2 z = 0 \
4 x - y + 3 z = 0
\end{array} \right)$$
A) (0, 0, 0)
B) inconsistent
C) dependent

Question

Solve the homogeneous system. $$\left( \begin{array} { r } 
x - 2 y + 5 z = 0 \
3 x + y - 2 z = 0 \
4 x - y + 3 z = 0
\end{array} \right)$$
A) (0, 0, 0)
B) inconsistent
C) dependent

Quizplus · Accepted Answer

We can find that the determinant of the coefficient matrix is 0. Therefore, the system has either infinitely many solutions or no solution. To see which one, we can row reduce the augmented matrix $$\left( \begin{array} { r r r | r } 
1 & -2 & 5 & 0 \
3 & 1 & -2 & 0 \
4 & -1 & 3 & 0
\end{array} \right)$$ which gives us $$\left( \begin{array} { r r r | r } 
1 & 0 & 1 & 0 \
0 & 1 & 2 & 0 \
0 & 0 & 0 & 0
\end{array} \right).$$ Since the last row is all zeros, we have a dependent system. Therefore, the correct answer is C.

Solve the Homogeneous System (x−2y+5z=03x+y−2z=04x−y+3z=0)\left( \begin{array} { r } x - 2 y + 5 z = 0 \\3 x + y - 2 z = 0 \\4 x - y + 3 z = 0\end{array} \right)​x−2y+5z=03x+y−2z=04x−y+3z=0​​

Solve the Homogeneous System $\left( \begin{array} { r } x - 2 y + 5 z = 0 \\3 x + y - 2 z = 0 \\4 x - y + 3 z = 0\end{array} \right)$