Consider the following system of equations in matrix form 12λ12λxyz=0. Then which one of the following statements is true?

(A) ∀λ∈-∞, ∞, the given system has non-trivial solution

(B) ∀λ∈-∞, ∞, the given system has only trivial solution

(C) For λ≠0, the given system does not have any solution

(D) For λ=0, the given system is inconsistent

TS EAMCET · Maths · Determinants

Consider the following system of equations in matrix form $[\begin{matrix} 1 \\ 2 \\ λ \end{matrix}] [\begin{matrix} 1 & 2 & λ \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = 0$ . Then which one of the following statements is true?

A $\forall λ \in (- \infty, \infty),$ the given system has non-trivial solution
B $\forall λ \in (- \infty, \infty),$ the given system has only trivial solution
C For $λ \neq 0,$ the given system does not have any solution
D For $λ = 0,$ the given system is inconsistent

Verified Solution

Answer & Solution

Correct Answer

(A) $\forall λ \in (- \infty, \infty),$ the given system has non-trivial solution

Step-by-step Solution

Detailed explanation

Given system of equations 12λ12λxyz=0 ⇒1×11×21×λ2×12×22×λλ×1λ×2λ×λxyz=0 ⇒12λ242λλ2λλ2xyz=0 Now,…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Explore more questions on app

From TS EAMCET

Explore more questions on app