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4.2.1: Ejercicios 4.2

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    En Ejercicios\(\PageIndex{1}\) —\(\PageIndex{6}\),\(A\) se da una matriz. Para cada uno,

    1. Encontrar los valores propios de\(A\), y para cada valor propio, encontrar un autovector.
    2. Haz lo mismo para\(A^{T}\).
    3. Haz lo mismo para\(A^{−1}\)
    4. Encuentra\(\text{tr}(A)\).
    5. Encuentra\(\text{det}(A)\). Usa el Teorema 4.2.1 para verificar tus resultados.
    Ejercicio\(\PageIndex{1}\)

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

    Contestar
    1. \(\lambda_{1}=1\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{4}\\{1}\end{array}\right];\)
      \(\lambda_{2}=4\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{1}\\{1}\end{array}\right]\)
    2. \(\lambda_{1}=1\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{1}\end{array}\right];\)
      \(\lambda_{2}=4\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-1}\\{4}\end{array}\right]\)
    3. \(\lambda_{1}=1/4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{1}\\{1}\end{array}\right];\)
      \(\lambda_{2}=1\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{4}\\{1}\end{array}\right]\)
    4. \(5\)
    5. \(4\)
    Ejercicio\(\PageIndex{2}\)

    \(\left[\begin{array}{cc}{-2}&{-14}\\{-1}&{3}\end{array}\right]\)

    Contestar
    1. \(\lambda_{1}=-4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{7}\\{1}\end{array}\right];\)
      \(\lambda_{2}=5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-2}\\{1}\end{array}\right]\)
    2. \(\lambda_{1}=-4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{1}\\{2}\end{array}\right];\)
      \(\lambda_{2}=5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-1}\\{7}\end{array}\right]\)
    3. \(\lambda_{1}=-1/4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{7}\\{1}\end{array}\right];\)
      \(\lambda_{2}=1/5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-2}\\{1}\end{array}\right]\)
    4. \(1\)
    5. \(-20\)
    Ejercicio\(\PageIndex{3}\)

    \(\left[\begin{array}{cc}{5}&{30}\\{-1}&{-6}\end{array}\right]\)

    Contestar
    1. \(\lambda_{1}=-1\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-5}\\{1}\end{array}\right];\)
      \(\lambda_{2}=0\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-6}\\{1}\end{array}\right]\)
    2. \(\lambda_{1}=-1\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{1}\\{6}\end{array}\right];\)
      \(\lambda_{2}=0\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{1}\\{5}\end{array}\right]\)
    3. \(A\)no es invertible.
    4. \(-1\)
    5. \(0\)
    Ejercicio\(\PageIndex{4}\)

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

    Contestar
    1. \(\lambda_{1}=4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{9}\\{1}\end{array}\right];\)
      \(\lambda_{2}=5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{8}\\{1}\end{array}\right]\)
    2. \(\lambda_{1}=4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{8}\end{array}\right];\)
      \(\lambda_{2}=5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-1}\\{9}\end{array}\right]\)
    3. \(\lambda_{1}=1/4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{9}\\{1}\end{array}\right];\)
      \(\lambda_{2}=1/5\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{8}\\{1}\end{array}\right]\)
    4. \(9\)
    5. \(20\)
    Ejercicio\(\PageIndex{5}\)

    \(\left[\begin{array}{ccc}{5}&{-9}&{0}\\{1}&{-5}&{0}\\{2}&{4}&{3}\end{array}\right]\)

    Contestar
    1. \(\lambda_{1}=-4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-7}\\{-7}\\{6}\end{array}\right];\)
      \(\lambda_{2}=3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]\)
      \(\lambda_{3}=4\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{9}\\{1}\\{22}\end{array}\right]\)
    2. \(\lambda_{1}=-4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{9}\\{0}\end{array}\right];\)
      \(\lambda_{2}=3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{-20}\\{26}\\{7}\end{array}\right]\)
      \(\lambda_{3}=4\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]\)
    3. \(\lambda_{1}=-1/4\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-7}\\{-7}\\{6}\end{array}\right];\)
      \(\lambda_{2}=1/3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]\)
      \(\lambda_{3}=1/4\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{9}\\{1}\\{22}\end{array}\right]\)
    4. \(3\)
    5. \(-48\)
    Ejercicio\(\PageIndex{6}\)

    \(\left[\begin{array}{ccc}{0}&{25}&{0}\\{1}&{0}&{0}\\{1}&{1}&{-3}\end{array}\right]\)

    Contestar
    1. \(\lambda_{1}=-5\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-5}\\{1}\\{2}\end{array}\right];\)
      \(\lambda_{2}=-3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]\)
      \(\lambda_{3}=5\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{20}\\{4}\\{3}\end{array}\right]\)
    2. \(\lambda_{1}=-5\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{5}\\{0}\end{array}\right];\)
      \(\lambda_{2}=-3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{1}\\{-11}\\{8}\end{array}\right]\)
      \(\lambda_{3}=5\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{1}\\{5}\\{0}\end{array}\right]\)
    3. \(\lambda_{1}=-1/5\text{ with }\vec{x_{1}}=\left[\begin{array}{c}{-5}\\{1}\\{2}\end{array}\right];\)
      \(\lambda_{2}=-1/3\text{ with }\vec{x_{2}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]\)
      \(\lambda_{3}=1/5\text{ with }\vec{x_{3}}=\left[\begin{array}{c}{20}\\{4}\\{3}\end{array}\right]\)
    4. \(-3\)
    5. \(75\)

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