Question

2x − y + z = 0
3x + 2y − z = 0
x + 4y + 3z = 0

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\text{The given system of homogeneous equations can be written in matrix form as follows:} \\ \left[\begin{array}{ccc}2& -1& 1\\ 3& 2& -1\\ 1& 4& 3\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right] \\ \mathrm{or},AX=O \\ \mathrm{where},A=\left[\begin{array}{ccc}2& -1& 1\\ 3& 2& -1\\ 1& 4& 3\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\mathrm{and}O=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right] \\ \left|A\right|=\left|\begin{array}{ccc}2& -1& 1\\ 3& 2& -1\\ 1& 4& 3\end{array}\right| \\ =2\left(6+4\right)+1\left(9+1\right)+1(12-2) \\ =40 \\ \therefore \left|A\right|\text{≠0} \\ \text{So, the given system has only trivial solution, which is given below:} \\ \mathit{\text{x}}\text{=}\mathit{\text{y}}\text{=}\mathit{\text{z}}\text{=0} \end{gathered}$$