Question

The system of linear equations:
x + y + z = 2
2x + y − z = 3
3x + 2y + kz = 4 has a unique solution if
(a) k ≠ 0
(b) −1 < k < 1
(c) −2 < k < 2
(d) k = 0

Answer 1

(a) k ≠ 0

$$\begin{gathered} \text{For a unique solution,}\left|A\right|\text{≠ 0.} \\ \text{The given system of equations can be written in matrix form as follows:} \\ \left[\begin{array}{ccc}1& 1& 1\\ 2& 1& -1\\ 3& 2& k\end{array}\right]\text{≠0} \\ \Rightarrow 1\left(k+2\right)-1\left(2k+3\right)+1\left(4-3\right)\text{≠0} \\ \Rightarrow k+2-2k-3+1\text{≠ 0} \\ \Rightarrow k\ne 0 \\ \text{So, the given system of equations has a unique solution if}\mathit{\text{k}}\text{is not equal to 0.} \end{gathered}$$