Question

The system of equation x + y + z = 2, 3x − y + 2z = 6 and 3x + y + z = −18 has

Options

• a unique solution

• no solution

• an infinite number of solutions

• zero solution as the only solution

Answer 1

(a) a unique solution

The given system of equations can be written in matrix form as follows:
\[\begin{bmatrix}1 & 1 & 1 \\ 3 & - 1 & 2 \\ 3 & 1 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix} = \begin{bmatrix}2 \\ 6 \\ - 18\end{bmatrix}\]
\[AX = B \]
Here, 
\[A = \begin{bmatrix}1 & 1 & 1 \\ 3 & - 1 & 2 \\ 3 & 1 & 1\end{bmatrix}, X = \begin{bmatrix}x \\ y \\ z\end{bmatrix}\text{ and }B = \begin{bmatrix}2 \\ 6 \\ - 18\end{bmatrix}\]
\[\left| A \right|=1 \left( - 1 - 2 \right) - 1\left( 3 - 6 \right) + 1\left( 3 + 3 \right)\]
\[ = - 3 + 3 + 6\]
\[ = 6\neq0\]
So, the given system of equations has a unique solution.