Matrices & Determinants
Consistency of Linear System
nta_pyq_2025_apr
Grade 12
Question:
The system of equations $x + y + z = 6$, $x + 2y + 5z = 9$, $x + 5y + \lambda z = \mu$ has no solution if:
$\lambda = 15, \mu \neq 17$
$\lambda \neq 17, \mu \neq 18$
$\lambda = 17, \mu \neq 18$
$\lambda = 17, \mu = 18$
Step-by-Step Solution
Key Concept: For no solution: the coefficient determinant $D = 0$ (so $\lambda = 17$) but at least one of $D_x, D_y, D_z \neq 0$ (so $\mu \neq 18$).
$D = \begin{vmatrix}1&1&1\\1&2&5\\1&5&\lambda\end{vmatrix} = 0 \Rightarrow \lambda = 17$. With $\lambda = 17$: $D_z = \begin{vmatrix}1&1&6\\1&2&9\\1&5&\mu\end{vmatrix} \neq 0 \Rightarrow \mu \neq 18$. So no solution iff $\lambda = 17, \mu \neq 18$.
Correct Answer: $\lambda = 17, \mu \neq 18$