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Mathematics LibreTexts

4.1.1: Exercises 4.1

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    Dec 19, 2024
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( \newcommand{\kernel}{\mathrm{null}\,}\)

In Exercises 4.1.1.1 - 4.1.1.6, a matrix A and one of its eigenvectors are given. Find the eigenvalue of A for the given eigenvector.

Exercise 4.1.1.1

A=[98−6−5]→x=[−43]

Answer

λ=3

Exercise 4.1.1.2

A=[19−648−15]→x=[13]

Answer

λ=1

Exercise 4.1.1.3

A=[1−2−24]→x=[21]

Answer

λ=0

Exercise 4.1.1.4

A=[−11−1914−6−86−12−2215]→x=[324]

Answer

λ=−5

Exercise 4.1.1.5

A=[−713102−3−20−141]→x=[1−24]

Answer

λ=3

Exercise 4.1.1.6

A=[−12−100151301518−5]→x=[−111]

Answer

λ=−2

In Exercises 4.1.1.7 – 4.1.1.11, a matrix A and one of its eigenvalues are given. Find an eigenvector of A for the given eigenvalue.

Exercise 4.1.1.7

A=[166−18−5]λ=4

Answer

→x=[−12]

Exercise 4.1.1.8

A=[−26−913]λ=7

Answer

→x=[23]

Exercise 4.1.1.9

A=[−16−28−19426946−42−72−49]λ=5

Answer

→x=[3−77]

Exercise 4.1.1.10

A=[7−5−1062−62−5−5]λ=−3

Answer

→x=[101]

Exercise 4.1.1.11

A=[45−3−7−831−58]λ=2

Answer

→x=[−111]

In Exercises 4.1.1.12 – 4.1.1.28, find the eigenvalues of the given matrix. For each eigenvalue, give an eigenvector.

Exercise 4.1.1.12

[−1−4−3−2]

Answer

λ1=−5 with →x1=[11];

λ2=2 with →x2=[−43]

Exercise 4.1.1.13

[−472−113]

Answer

λ1=4 with →x1=[91];

λ2=5 with →x2=[81]

Exercise 4.1.1.14

[2−122−8]

Answer

λ1=−4 with →x1=[21];

λ2=−2 with →x2=[31]

Exercise 4.1.1.15

[3121−1]

Answer

λ1=−3 with →x1=[−21];

λ2=5 with →x2=[61]

Exercise 4.1.1.16

[59−1−5]

Answer

λ1=−4 with →x1=[−11];

λ2=4 with →x2=[−91]

Exercise 4.1.1.17

[3−1−13]

Answer

λ1=2 with →x1=[11];

λ2=4 with →x2=[−11]

Exercise 4.1.1.18

[01250]

Answer

λ1=−5 with →x1=[−15];

λ2=5 with →x2=[15]

Exercise 4.1.1.19

[−310−1]

Answer

λ1=−1 with →x1=[12];

λ2=−3 with →x2=[10]

Exercise 4.1.1.20

[1−2−30300−1−1]

Answer

λ1=−1 with →x1=[302];

λ2=1 with →x2=[100]

λ3=3 with →x3=[5−82]

Exercise 4.1.1.21

[5−230400−13]

Answer

λ1=3 with →x1=[−302];

λ2=4 with →x2=[−5−11]

λ3=5 with →x3=[100]

Exercise 4.1.1.22

[10122−50102]

Answer

λ1=−5 with →x1=[010];

λ2=−2 with →x2=[−12−83]

λ3=5 with →x3=[1535]

Exercise 4.1.1.23

[10−18−43−110−8]

Answer

λ1=−5 with →x1=[24138];

λ2=−2 with →x2=[651]

λ3=3 with →x3=[010]

Exercise 4.1.1.24

[−11801205−3−1]

Answer

λ1=−4 with →x1=[−6111];

λ2=−1 with →x2=[001]

λ3=5 with →x3=[312]

Exercise 4.1.1.25

[500110−15−2]

Answer

λ1=−2 with →x1=[001];

λ2=1 with →x2=[035]

λ3=5 with →x3=[2871]

Exercise 4.1.1.26

[2−11036007]

Answer

λ1=2 with →x1=[100];

λ2=3 with →x2=[−110]

λ3=7 with →x3=[−11510]

Exercise 4.1.1.27

[35−5−232−250]

Answer

λ1=−2 with →x1=[101];

λ2=3 with →x2=[111];

λ3=5 with →x3=[011]

This page titled 4.1.1: Exercises 4.1 is shared under a CC BY-NC 3.0 license and was authored, remixed, and/or curated by Gregory Hartman et al. via source content that was edited to the style and standards of the LibreTexts platform.

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