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Questions › IIT JAM › IIT JAM 2022 › Q47
1 markMathematics › Matrices, determinants and linear systems › Consistency, rank and solution spacesModerateabout 2 min

IIT JAM 2022 Question 47: Matrices, determinants and linear systems

Let the system be αu+w=0\alpha u + w = 0αu+w=0, u+αv=0u + \alpha v = 0u+αv=0, v+αw=0v + \alpha w = 0v+αw=0, where α∈R\alpha \in \mathbb{R}α∈R. The system has infinite solutions if α=\alpha = α= _______ (in integer).

Answer and worked solution

Answer: -1 to -1

The system is homogeneous, so it has nontrivial (infinitely many) solutions exactly when the coefficient determinant vanishes.

With variables (u,v,w)(u, v, w)(u,v,w) the coefficient matrix has determinant α3+1\alpha^3 + 1α3+1. Setting α3+1=0\alpha^3 + 1 = 0α3+1=0 gives the real root α=−1\alpha = -1α=−1. The answer is −1-1−1.

Plain-text version

Let the system be α u + w = 0, u + α v = 0, v + α w = 0, where α ∈ ℝ. The system has infinite solutions if α = _ (in integer).

Also written as: Let the system be alpha u + w = 0, u + alpha v = 0, v + alpha w = 0, where alpha in R. The system has infinite solutions if alpha = _ (in integer).

Matrices, determinants and linear systems has come up 102 times in 7 exams since 2006.

What this question tests

This question is on Consistency, rank and solution spaces, part of Matrices, determinants and linear systems in Mathematics.

Whether a system has no solution, one or many, and the shape of its solution set.

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Question from the IIT JAM 2022 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.