4.1.1: Ejercicios 4.1

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En Ejercicios\(\PageIndex{1}\) -\(\PageIndex{6}\), se dan una matriz\(A\) y uno de sus vectores propios. Encuentra el valor propio de\(A\) para el vector propio dado.

Ejercicio\(\PageIndex{1}\)

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

Contestar: \(\lambda =3\)

Ejercicio\(\PageIndex{2}\)

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

Contestar: \(\lambda =1\)

Ejercicio\(\PageIndex{3}\)

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

Contestar: \(\lambda =0\)

Ejercicio\(\PageIndex{4}\)

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

Contestar: \(\lambda =-5\)

Ejercicio\(\PageIndex{5}\)

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

Contestar: \(\lambda =3\)

Ejercicio\(\PageIndex{6}\)

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

Contestar: \(\lambda =-2\)

En Ejercicios\(\PageIndex{7}\) —\(\PageIndex{11}\), se dan una matriz\(A\) y uno de sus valores propios. Encuentra un vector propio de\(A\) para el valor propio dado.

Ejercicio\(\PageIndex{7}\)

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

Contestar: \(\vec{x}=\left[\begin{array}{c}{-1}\\{2}\end{array}\right]\)

Ejercicio\(\PageIndex{8}\)

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

Contestar: \(\vec{x}=\left[\begin{array}{c}{2}\\{3}\end{array}\right]\)

Ejercicio\(\PageIndex{9}\)

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

Contestar: \(\vec{x}=\left[\begin{array}{c}{3}\\{-7}\\{7}\end{array}\right]\)

Ejercicio\(\PageIndex{10}\)

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

Contestar: \(\vec{x}=\left[\begin{array}{c}{1}\\{0}\\{1}\end{array}\right]\)

Ejercicio\(\PageIndex{11}\)

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

Contestar: \(\vec{x}=\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]\)

En Ejercicios\(\PageIndex{12}\) —\(\PageIndex{28}\), encuentra los valores propios de la matriz dada. Para cada valor propio, dar un vector propio.

Ejercicio\(\PageIndex{12}\)

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

Contestar

\(\lambda_{1}=-5\)con\(\vec{x_{1}}=\left[\begin{array}{c}{1}\\{1}\end{array}\right];\)

\(\lambda_{2}=2\)con\(\vec{x_{2}}=\left[\begin{array}{c}{-4}\\{3}\end{array}\right]\)

Ejercicio\(\PageIndex{13}\)

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

Contestar

\(\lambda_{1}=4\)con\(\vec{x_{1}}=\left[\begin{array}{c}{9}\\{1}\end{array}\right];\)

\(\lambda_{2}=5\)con\(\vec{x_{2}}=\left[\begin{array}{c}{8}\\{1}\end{array}\right]\)

Ejercicio\(\PageIndex{14}\)

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

Contestar

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

\(\lambda_{2}=-2\)con\(\vec{x_{2}}=\left[\begin{array}{c}{3}\\{1}\end{array}\right]\)

Ejercicio\(\PageIndex{15}\)

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

Contestar

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

\(\lambda_{2}=5\)con\(\vec{x_{2}}=\left[\begin{array}{c}{6}\\{1}\end{array}\right]\)

Ejercicio\(\PageIndex{16}\)

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

Contestar

\(\lambda_{1}=-4\)con\(\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{1}\end{array}\right];\)

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

Ejercicio\(\PageIndex{17}\)

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

Contestar

\(\lambda_{1}=2\)con\(\vec{x_{1}}=\left[\begin{array}{c}{1}\\{1}\end{array}\right];\)

\(\lambda_{2}=4\)con\(\vec{x_{2}}=\left[\begin{array}{c}{-1}\\{1}\end{array}\right]\)

Ejercicio\(\PageIndex{18}\)

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

Contestar

\(\lambda_{1}=-5\)con\(\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{5}\end{array}\right];\)

\(\lambda_{2}=5\)con\(\vec{x_{2}}=\left[\begin{array}{c}{1}\\{5}\end{array}\right]\)

Ejercicio\(\PageIndex{19}\)

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

Contestar

\(\lambda_{1}=-1\)con\(\vec{x_{1}}=\left[\begin{array}{c}{1}\\{2}\end{array}\right];\)

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

Ejercicio\(\PageIndex{20}\)

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

Contestar

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

\(\lambda_{2}=1\)con\(\vec{x_{2}}=\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right]\)

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

Ejercicio\(\PageIndex{21}\)

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

Contestar

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

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

\(\lambda_{3}=5\)con\(\vec{x_{3}}=\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right]\)

Ejercicio\(\PageIndex{22}\)

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

Contestar

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

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

\(\lambda_{3}=5\)con\(\vec{x_{3}}=\left[\begin{array}{c}{15}\\{3}\\{5}\end{array}\right]\)

Ejercicio\(\PageIndex{23}\)

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

Contestar

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

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

\(\lambda_{3}=3\)con\(\vec{x_{3}}=\left[\begin{array}{c}{0}\\{1}\\{0}\end{array}\right]\)

Ejercicio\(\PageIndex{24}\)

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

Contestar

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

\(\lambda_{2}=-1\)con\(\vec{x_{2}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]\)

\(\lambda_{3}=5\)con\(\vec{x_{3}}=\left[\begin{array}{c}{3}\\{1}\\{2}\end{array}\right]\)

Ejercicio\(\PageIndex{25}\)

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

Contestar

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

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

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

Ejercicio\(\PageIndex{26}\)

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

Contestar

\(\lambda_{1}=2\)con\(\vec{x_{1}}=\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right];\)

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

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

Ejercicio\(\PageIndex{27}\)

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

Contestar

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

\(\lambda_{2}=3\)con\(\vec{x_{2}}=\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right];\)

\(\lambda_{3}=5\)con\(\vec{x_{3}}=\left[\begin{array}{c}{0}\\{1}\\{1}\end{array}\right]\)

Ejercicio\(\PageIndex{28}\)

\(\left[\begin{array}{ccc}{1}&{2}&{1}\\{1}&{2}&{3}\\{1}&{1}&{1}\end{array}\right]\)

Contestar

\(\lambda_{1}=0\)con\(\vec{x_{1}}=\left[\begin{array}{c}{1}\\{3}\\{1}\end{array}\right];\)

\(\lambda_{2}=-1\)con\(\vec{x_{2}}=\left[\begin{array}{c}{2}\\{2}\\{1}\end{array}\right];\)

\(\lambda_{3}=2\)con\(\vec{x_{3}}=\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right]\)