Which of the following is/are the eigenvector(s) for the matrix given below? pmatrix -9 & -6 & -2 & -4 \\ -8 & -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32 & 21 & 7 & 12 pmatrix

pmatrix -1 \\ 1 \\ 0 \\ 1 pmatrix

pmatrix -1 \\ 0 \\ 2 \\ 2 pmatrix

pmatrix 0 \\ 1 \\ -3 \\ 0 pmatrix

pmatrix -1 \\ 1 \\ 0 \\ 1 pmatrix

pmatrix 1 \\ 0 \\ -1 \\ 0 pmatrix

pmatrix -1 \\ 0 \\ 2 \\ 2 pmatrix

pmatrix 0 \\ 1 \\ -3 \\ 0 pmatrix

GATE CSE 2022 Q53: Which vectors are eigenvectors of a 4×4 matrix

Which of the following is/are the eigenvector(s) for the matrix given below?

$\begin{pmatrix} -9 & -6 & -2 & -4 \\ -8 & -6 & -3 & -1 \\ 20 & 15 & 8 & 5 \\ 32 & 21 & 7 & 12 \end{pmatrix}$

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Answer: A, C, D (official GATE 2022 final answer key)

Concept used: $v$ is an eigenvector exactly when $Mv$ is a scalar multiple of $v$ .

First step: Multiply the matrix by each vector; for (A), $Mv = (-1, 1, 0, 1)^T$ , so it is an eigenvector with eigenvalue 1.

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Source: official GATE 2022 CS question paper and final answer key, organised by IIT Kharagpur. Question paper · Answer key