Question

The system of equations:
x + y + z = 5
x + 2y + 3z = 9
x + 3y + λz = µ
has a unique solution, if
(a) λ = 5, µ = 13
(b) λ ≠ 5
(c) λ = 5, µ ≠ 13
(d) µ ≠ 13

Answer 1

$$\begin{gathered} \left(\mathrm{b}\right)\lambda \ne 5 \\ \text{For a unique solution,}\left|A\right|\text{≠ 0.} \\ \Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& \lambda \end{array}\right|\ne 0 \\ \Rightarrow 1\left(2\lambda -9\right)-1\left(\lambda -3\right)+1\left(3-2\right)\ne 0 \\ \Rightarrow 2\lambda -9-\lambda +3+1\ne 0 \\ \Rightarrow \lambda -5\ne 0 \\ \Rightarrow \lambda \ne 5 \end{gathered}$$