Question

3x + y − 2z = 0
x + y + z = 0
x − 2y + z = 0

Answer 1

$$\begin{gathered} \text{The given system of homogeneous equations can be written in matrix form as follows:} \\ \left[\begin{array}{ccc}3& 1& -2\\ 1& 1& 1\\ 1& -2& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right] \\ \mathrm{or},AX=O \\ \mathrm{where},A=\left[\begin{array}{ccc}3& 1& -2\\ 1& 1& 1\\ 1& -2& 1\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\mathrm{and}O=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right] \\ \mathrm{Now}, \\ \left|A\right|=\left|\begin{array}{ccc}3& 1& -2\\ 1& 1& 1\\ 1& -2& 1\end{array}\right| \\ =3\left(1+2\right)-1\left(1-1\right)-2\left(-2-1\right) \\ =9-0+6 \\ =15\ne 0 \\ \text{So, the given system has only trivial solution, which is given below:} \\ \mathit{\text{x}}\text{=}\mathit{\text{y}}\text{=}\mathit{\text{z}}\text{=0} \end{gathered}$$