Question

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

Answer 1

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