Question

The system of equation x + y + z = 2, 3x − y + 2z = 6 and 3x + y + z = −18 has
(a) a unique solution
(b) no solution
(c) an infinite number of solutions
(d) zero solution as the only solution

Answer 1

(a) a unique solution

$$\begin{gathered} \text{The given system of equations can be written in matrix form as follows:} \\ \left[\begin{array}{ccc}1& 1& 1\\ 3& -1& 2\\ 3& 1& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}2\\ 6\\ -18\end{array}\right] \\ AX=B \\ \mathrm{Here}, \\ A=\left[\begin{array}{ccc}1& 1& 1\\ 3& -1& 2\\ 3& 1& 1\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]\mathrm{and}B=\left[\begin{array}{c}2\\ 6\\ -18\end{array}\right] \\ \left|A\right|\text{=}1\left(-1-2\right)-1\left(3-6\right)+1\left(3+3\right) \\ =-3+3+6 \\ =6\text{≠0} \\ \text{So, the given system of equations has a unique solution.} \end{gathered}$$