**Question:**

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

(a) a unique solution

(b) no solution

(c) an infinite number of solutions

(d) zero solution as the only solution

**Solution:**

(a) a unique solution

The given system of equations can be written in matrix form as follows:

$\left[\begin{array}{ccc}1 & 1 & 1 \\ 3 & -1 & 2 \\ 3 & 1 & 1\end{array}\right]\left[\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[\begin{array}{c}2 \\ 6 \\ -18\end{array}\right]$

$A X=B$

Here,

$A=\left[\begin{array}{ccc}1 & 1 & 1 \\ 3 & -1 & 2 \\ 3 & 1 & 1\end{array}\right], X=\left[\begin{array}{c}x \\ y \\ z\end{array}\right]$ and $B=\left[\begin{array}{c}2 \\ 6 \\ -18\end{array}\right]$

$|A|=1(-1-2)-1(3-6)+1(3+3)$

$=-3+3+6$

$=6 \neq 0$

So, the given system of equations has a unique solution.