4.1.1: Exercises 4.1

In Exercises $\PageIndex{1}$ - $\PageIndex{6}$, a matrix $A$ and one of its eigenvectors are given. Find the eigenvalue of $A$ for the given eigenvector.

Exercise $\PageIndex{1}$

$A=\left[\begin{array}{cc}{9}&{8}\\{-6}&{-5}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{-4}\\{3}\end{array}\right]$

Answer

$\lambda =3$

Exercise $\PageIndex{2}$

$A=\left[\begin{array}{cc}{19}&{-6}\\{48}&{-15}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{1}\\{3}\end{array}\right]$

Answer

$\lambda =1$

Exercise $\PageIndex{3}$

$A=\left[\begin{array}{cc}{1}&{-2}\\{-2}&{4}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{2}\\{1}\end{array}\right]$

Answer

$\lambda =0$

Exercise $\PageIndex{4}$

$A=\left[\begin{array}{ccc}{-11}&{-19}&{14}\\{-6}&{-8}&{6}\\{-12}&{-22}&{15}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{3}\\{2}\\{4}\end{array}\right]$

Answer

$\lambda =-5$

Exercise $\PageIndex{5}$

$A=\left[\begin{array}{ccc}{-7}&{1}&{3}\\{10}&{2}&{-3}\\{-20}&{-14}&{1}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{1}\\{-2}\\{4}\end{array}\right]$

Answer

$\lambda =3$

Exercise $\PageIndex{6}$

$A=\left[\begin{array}{ccc}{-12}&{-10}&{0}\\{15}&{13}&{0}\\{15}&{18}&{-5}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]$

Answer

$\lambda =-2$

In Exercises $\PageIndex{7}$ – $\PageIndex{11}$, a matrix $A$ and one of its eigenvalues are given. Find an eigenvector of $A$ for the given eigenvalue.

Exercise $\PageIndex{7}$

$A=\left[\begin{array}{cc}{16}&{6}\\{-18}&{-5}\end{array}\right]\quad\lambda =4$

Answer

$\vec{x}=\left[\begin{array}{c}{-1}\\{2}\end{array}\right]$

Exercise $\PageIndex{8}$

$A=\left[\begin{array}{cc}{-2}&{6}\\{-9}&{13}\end{array}\right]\quad\lambda =7$

Answer

$\vec{x}=\left[\begin{array}{c}{2}\\{3}\end{array}\right]$

Exercise $\PageIndex{9}$

$A=\left[\begin{array}{ccc}{-16}&{-28}&{-19}\\{42}&{69}&{46}\\{-42}&{-72}&{-49}\end{array}\right]\quad\lambda =5$

Answer

$\vec{x}=\left[\begin{array}{c}{3}\\{-7}\\{7}\end{array}\right]$

Exercise $\PageIndex{10}$

$A=\left[\begin{array}{ccc}{7}&{-5}&{-10}\\{6}&{2}&{-6}\\{2}&{-5}&{-5}\end{array}\right]\quad\lambda =-3$

Answer

$\vec{x}=\left[\begin{array}{c}{1}\\{0}\\{1}\end{array}\right]$

Exercise $\PageIndex{11}$

$A=\left[\begin{array}{ccc}{4}&{5}&{-3}\\{-7}&{-8}&{3}\\{1}&{-5}&{8}\end{array}\right]\quad\lambda =2$

Answer

$\vec{x}=\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]$

In Exercises $\PageIndex{12}$ – $\PageIndex{28}$, find the eigenvalues of the given matrix. For each eigenvalue, give an eigenvector.

Exercise $\PageIndex{12}$

$\left[\begin{array}{cc}{-1}&{-4}\\{-3}&{-2}\end{array}\right]$

Answer

$\lambda_{1}=-5$ with $\vec{x_{1}}=\left[\begin{array}{c}{1}\\{1}\end{array}\right];$

$\lambda_{2}=2$ with $\vec{x_{2}}=\left[\begin{array}{c}{-4}\\{3}\end{array}\right]$

Exercise $\PageIndex{13}$

$\left[\begin{array}{cc}{-4}&{72}\\{-1}&{13}\end{array}\right]$

Answer

$\lambda_{1}=4$ with $\vec{x_{1}}=\left[\begin{array}{c}{9}\\{1}\end{array}\right];$

$\lambda_{2}=5$ with $\vec{x_{2}}=\left[\begin{array}{c}{8}\\{1}\end{array}\right]$

Exercise $\PageIndex{14}$

$\left[\begin{array}{cc}{2}&{-12}\\{2}&{-8}\end{array}\right]$

Answer

$\lambda_{1}=-4$ with $\vec{x_{1}}=\left[\begin{array}{c}{2}\\{1}\end{array}\right];$

$\lambda_{2}=-2$ with $\vec{x_{2}}=\left[\begin{array}{c}{3}\\{1}\end{array}\right]$

Exercise $\PageIndex{15}$

$\left[\begin{array}{cc}{3}&{12}\\{1}&{-1}\end{array}\right]$

Answer

$\lambda_{1}=-3$ with $\vec{x_{1}}=\left[\begin{array}{c}{-2}\\{1}\end{array}\right];$

$\lambda_{2}=5$ with $\vec{x_{2}}=\left[\begin{array}{c}{6}\\{1}\end{array}\right]$

Exercise $\PageIndex{16}$

$\left[\begin{array}{cc}{5}&{9}\\{-1}&{-5}\end{array}\right]$

Answer

$\lambda_{1}=-4$ with $\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{1}\end{array}\right];$

$\lambda_{2}=4$ with $\vec{x_{2}}=\left[\begin{array}{c}{-9}\\{1}\end{array}\right]$

Exercise $\PageIndex{17}$

$\left[\begin{array}{cc}{3}&{-1}\\{-1}&{3}\end{array}\right]$

Answer

$\lambda_{1}=2$ with $\vec{x_{1}}=\left[\begin{array}{c}{1}\\{1}\end{array}\right];$

$\lambda_{2}=4$ with $\vec{x_{2}}=\left[\begin{array}{c}{-1}\\{1}\end{array}\right]$

Exercise $\PageIndex{18}$

$\left[\begin{array}{cc}{0}&{1}\\{25}&{0}\end{array}\right]$

Answer

$\lambda_{1}=-5$ with $\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{5}\end{array}\right];$

$\lambda_{2}=5$ with $\vec{x_{2}}=\left[\begin{array}{c}{1}\\{5}\end{array}\right]$

Exercise $\PageIndex{19}$

$\left[\begin{array}{cc}{-3}&{1}\\{0}&{-1}\end{array}\right]$

Answer

$\lambda_{1}=-1$ with $\vec{x_{1}}=\left[\begin{array}{c}{1}\\{2}\end{array}\right];$

$\lambda_{2}=-3$ with $\vec{x_{2}}=\left[\begin{array}{c}{1}\\{0}\end{array}\right]$

Exercise $\PageIndex{20}$

$\left[\begin{array}{ccc}{1}&{-2}&{-3}\\{0}&{3}&{0}\\{0}&{-1}&{-1}\end{array}\right]$

Answer

$\lambda_{1}=-1$ with $\vec{x_{1}}=\left[\begin{array}{c}{3}\\{0}\\{2}\end{array}\right];$

$\lambda_{2}=1$ with $\vec{x_{2}}=\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right]$

$\lambda_{3}=3$ with $\vec{x_{3}}=\left[\begin{array}{c}{5}\\{-8}\\{2}\end{array}\right]$

Exercise $\PageIndex{21}$

$\left[\begin{array}{ccc}{5}&{-2}&{3}\\{0}&{4}&{0}\\{0}&{-1}&{3}\end{array}\right]$

Answer

$\lambda_{1}=3$ with $\vec{x_{1}}=\left[\begin{array}{c}{-3}\\{0}\\{2}\end{array}\right];$

$\lambda_{2}=4$ with $\vec{x_{2}}=\left[\begin{array}{c}{-5}\\{-1}\\{1}\end{array}\right]$

$\lambda_{3}=5$ with $\vec{x_{3}}=\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right]$

Exercise $\PageIndex{22}$

$\left[\begin{array}{ccc}{1}&{0}&{12}\\{2}&{-5}&{0}\\{1}&{0}&{2}\end{array}\right]$

Answer

$\lambda_{1}=-5$ with $\vec{x_{1}}=\left[\begin{array}{c}{0}\\{1}\\{0}\end{array}\right];$

$\lambda_{2}=-2$ with $\vec{x_{2}}=\left[\begin{array}{c}{-12}\\{-8}\\{3}\end{array}\right]$

$\lambda_{3}=5$ with $\vec{x_{3}}=\left[\begin{array}{c}{15}\\{3}\\{5}\end{array}\right]$

Exercise $\PageIndex{23}$

$\left[\begin{array}{ccc}{1}&{0}&{-18}\\{-4}&{3}&{-1}\\{1}&{0}&{-8}\end{array}\right]$

Answer

$\lambda_{1}=-5$ with $\vec{x_{1}}=\left[\begin{array}{c}{24}\\{13}\\{8}\end{array}\right];$

$\lambda_{2}=-2$ with $\vec{x_{2}}=\left[\begin{array}{c}{6}\\{5}\\{1}\end{array}\right]$

$\lambda_{3}=3$ with $\vec{x_{3}}=\left[\begin{array}{c}{0}\\{1}\\{0}\end{array}\right]$

Exercise $\PageIndex{24}$

$\left[\begin{array}{ccc}{-1}&{18}&{0}\\{1}&{2}&{0}\\{5}&{-3}&{-1}\end{array}\right]$

Answer

$\lambda_{1}=-4$ with $\vec{x_{1}}=\left[\begin{array}{c}{-6}\\{1}\\{11}\end{array}\right];$

$\lambda_{2}=-1$ with $\vec{x_{2}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]$

$\lambda_{3}=5$ with $\vec{x_{3}}=\left[\begin{array}{c}{3}\\{1}\\{2}\end{array}\right]$

Exercise $\PageIndex{25}$

$\left[\begin{array}{ccc}{5}&{0}&{0}\\{1}&{1}&{0}\\{-1}&{5}&{-2}\end{array}\right]$

Answer

$\lambda_{1}=-2$ with $\vec{x_{1}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right];$

$\lambda_{2}=1$ with $\vec{x_{2}}=\left[\begin{array}{c}{0}\\{3}\\{5}\end{array}\right]$

$\lambda_{3}=5$ with $\vec{x_{3}}=\left[\begin{array}{c}{28}\\{7}\\{1}\end{array}\right]$

Exercise $\PageIndex{26}$

$\left[\begin{array}{ccc}{2}&{-1}&{1}\\{0}&{3}&{6}\\{0}&{0}&{7}\end{array}\right]$

Answer

$\lambda_{1}=2$ with $\vec{x_{1}}=\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right];$

$\lambda_{2}=3$ with $\vec{x_{2}}=\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]$

$\lambda_{3}=7$ with $\vec{x_{3}}=\left[\begin{array}{c}{-1}\\{15}\\{10}\end{array}\right]$

Exercise $\PageIndex{27}$

$\left[\begin{array}{ccc}{3}&{5}&{-5}\\{-2}&{3}&{2}\\{-2}&{5}&{0}\end{array}\right]$

Answer

$\lambda_{1}=-2$ with $\vec{x_{1}}=\left[\begin{array}{c}{1}\\{0}\\{1}\end{array}\right];$

$\lambda_{2}=3$ with $\vec{x_{2}}=\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right];$

$\lambda_{3}=5$ with $\vec{x_{3}}=\left[\begin{array}{c}{0}\\{1}\\{1}\end{array}\right]$