Question

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

Answer 1

$$\begin{gathered} \text{The given system of homogeneous equations can be written in matrix form as follows:} \\ \left[\begin{array}{ccc}2& 3& -1\\ 1& -1& -2\\ 3& 1& 3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right] \\ AX=O \\ \mathrm{Here}, \\ A=\left[\begin{array}{ccc}2& 3& -1\\ 1& -1& -2\\ 3& 1& 3\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\mathrm{and}O=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right] \\ \mathrm{Now}, \\ \left|A\right|=\left|\begin{array}{ccc}2& 3& -1\\ 1& -1& -2\\ 3& 1& 3\end{array}\right| \\ =2\left(-3+2\right)-3\left(3+6\right)-1\left(1+3\right) \\ =-2-27-4 \\ =-33\ne 0 \\ \text{So, the given system}\mathrm{of}\mathrm{homogeneous}equations\mathrm{h}\text{as only trivial solution, which is given below:} \\ \mathit{\text{x}}\text{=}\mathit{\text{y}}\text{=}\mathit{\text{z}}\text{=0} \end{gathered}$$