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

Options

• λ = 5, µ = 13

• λ ≠ 5

• λ = 5, µ ≠ 13

• µ ≠ 13

Answer 1

\[(b) \lambda \neq 5\]
\[\text{ For a unique solution,}\left| A \right|\neq 0.\]
\[ \Rightarrow \begin{vmatrix}1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & \lambda\end{vmatrix} \neq 0\]
\[ \Rightarrow 1\left( 2\lambda - 9 \right) - 1\left( \lambda - 3 \right) + 1\left( 3 - 2 \right) \neq 0\]
\[ \Rightarrow 2\lambda - 9 - \lambda + 3 + 1 \neq 0\]
\[ \Rightarrow \lambda - 5 \neq 0\]
\[ \Rightarrow \lambda \neq 5\]