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Questions › IIT JAM › IIT JAM 2023 › Q35
2 marksMathematics › Vector spaces, rank and eigenvalues › Eigenvalues and eigenvectorsHardabout 3 min

IIT JAM 2023 Question 35: Vector spaces, rank and eigenvalues

For α,β∈R\alpha,\beta\in\mathbb{R}α,β∈R, α≠β\alpha\ne\betaα=β, if −2-2−2 and 555 are eigenvalues of M=(1−α1+ββα+β)M=\begin{pmatrix}1-\alpha & 1+\beta\\ \beta & \alpha+\beta\end{pmatrix}M=(1−αβ​1+βα+β​) and X=[x1,x2]TX=[x_1,x_2]^TX=[x1​,x2​]T is an eigenvector for −2-2−2, then

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For α,β∈ℝ, α≠β, if -2 and 5 are eigenvalues of M=[1-α 1+β; β α+β] and X=[x₁,x₂]ᵀ is an eigenvector for -2, then (A) 2x₁+x₂=0 (B) β-α=5 (C) α²-β²=5 (D) x₁+3x₂=0

Also written as: For alpha,beta in R, alpha!=beta, if -2 and 5 are eigenvalues of M=[1-alpha 1+beta; beta alpha+beta] and X=[x_1,x_2]^T is an eigenvector for -2, then

Vector spaces, rank and eigenvalues has come up 41 times in 6 exams since 2008.

What this question tests

This question is on Eigenvalues and eigenvectors, part of Vector spaces, rank and eigenvalues in Mathematics.

The scalars and directions a matrix only stretches, and what they say about invertibility and trace.

TestStop has 6 past-paper questions on this subtopic; the nearest in year are below.

  • IIT JAM 2022 · Q41The sum of the eigenvalues of the square matrix [1 1 3; 1 5 1; 3 1 1] is _ (in integer).
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