determinats and matrices

Question

The system of linear equations x + y - z = 6, x + 2y - 3z = 14 and 2x + 5y - λz = 9 has a unique solution 

(A) λ = 8

(B) λ ≠ 8

(C) λ = 7

(D) λ ≠ 7

Answer 1

Correct option is (b) λ ≠ 8

For a system of linear equations having unique solution, value of the determinant should not be 0.

Here, \(\Delta = \begin{vmatrix} 1&1&-1\\1&2&-3\\2&5&-\lambda \end{vmatrix}\ne 0\)

\(\Rightarrow [-2\lambda + 15 - 1(-\lambda + 6) - (5-4)] \ne 0\)

\(\Rightarrow -\lambda + 8 \ne 0\)

\(\Rightarrow \lambda \ne 8\)