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