Question

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

Answer 1

 The given system of homogeneous equations can be written in matrix form as follows:
\[\begin{bmatrix}2 & - 1 & 1 \\ 3 & 2 & - 1 \\ 1 & 4 & 3\end{bmatrix} \begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}\]
\[or, AX = O\]
\[\text{ where, }A = \begin{bmatrix}2 & - 1 & 1 \\ 3 & 2 & - 1 \\ 1 & 4 & 3\end{bmatrix}, X = \begin{bmatrix}x \\ y \\ z\end{bmatrix}\text{ and }O = \begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}\]
\[ \left| A \right| = \begin{vmatrix}2 & - 1 & 1 \\ 3 & 2 & - 1 \\ 1 & 4 & 3\end{vmatrix}\]
\[ = 2\left( 6 + 4 \right) + 1\left( 9 + 1 \right) + 1(12 - 2)\]
\[ = 40\]
\[ \therefore \left| A \right| \neq0\]
So, the given system has only trivial solution, which is given below:
\[x=y=z=0\]