Eigenvalues of Matrix with variables












0












$begingroup$


Im supposed to find eigenvalues of following matrix:



$begin{bmatrix}1 & -x^2+x-2 & 0\1& 2x-1 & x-1\ 0& 4 & 1end{bmatrix}$



My attempt was to solve it the "normal way" to solve det(A-$lambda$I) = 0



But since there is a variable in the matrix, this leads to a big cubic equation which i cannot solve easily, which looks like:
$$lambda^3 - lambda ^2 -2xlambda + 5lambda + x^2 lambda -xlambda +3x - x^2 -5 =0$$
I know that this equation will add up for $lambda =1 $ but other than that i have no idea



am I missing something? is there a different way to attempt this?



Thanks










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    0












    $begingroup$


    Im supposed to find eigenvalues of following matrix:



    $begin{bmatrix}1 & -x^2+x-2 & 0\1& 2x-1 & x-1\ 0& 4 & 1end{bmatrix}$



    My attempt was to solve it the "normal way" to solve det(A-$lambda$I) = 0



    But since there is a variable in the matrix, this leads to a big cubic equation which i cannot solve easily, which looks like:
    $$lambda^3 - lambda ^2 -2xlambda + 5lambda + x^2 lambda -xlambda +3x - x^2 -5 =0$$
    I know that this equation will add up for $lambda =1 $ but other than that i have no idea



    am I missing something? is there a different way to attempt this?



    Thanks










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Im supposed to find eigenvalues of following matrix:



      $begin{bmatrix}1 & -x^2+x-2 & 0\1& 2x-1 & x-1\ 0& 4 & 1end{bmatrix}$



      My attempt was to solve it the "normal way" to solve det(A-$lambda$I) = 0



      But since there is a variable in the matrix, this leads to a big cubic equation which i cannot solve easily, which looks like:
      $$lambda^3 - lambda ^2 -2xlambda + 5lambda + x^2 lambda -xlambda +3x - x^2 -5 =0$$
      I know that this equation will add up for $lambda =1 $ but other than that i have no idea



      am I missing something? is there a different way to attempt this?



      Thanks










      share|cite|improve this question









      $endgroup$




      Im supposed to find eigenvalues of following matrix:



      $begin{bmatrix}1 & -x^2+x-2 & 0\1& 2x-1 & x-1\ 0& 4 & 1end{bmatrix}$



      My attempt was to solve it the "normal way" to solve det(A-$lambda$I) = 0



      But since there is a variable in the matrix, this leads to a big cubic equation which i cannot solve easily, which looks like:
      $$lambda^3 - lambda ^2 -2xlambda + 5lambda + x^2 lambda -xlambda +3x - x^2 -5 =0$$
      I know that this equation will add up for $lambda =1 $ but other than that i have no idea



      am I missing something? is there a different way to attempt this?



      Thanks







      linear-algebra eigenvalues-eigenvectors






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      asked Jan 5 at 22:50









      Nicola ZauggNicola Zaugg

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          $begingroup$

          Since you know that $ lambda = 1$ works, note that your equation factors as
          $$(lambda - 1) (x^2 - 3x + 5 + lambda^2) = 0.$$
          This gives that
          $$lambda = 1, lambda = pm sqrt{3x - x^2 - 5}.$$






          share|cite|improve this answer









          $endgroup$





















            1












            $begingroup$

            Since you know that $1$ is a root of your polynomial, divide it by $lambda-1$, getting$$lambda^3-lambda^2-2xlambda+5lambda+x^2lambda-xlambda+3x-x^2-5=(lambda-1)(lambda^2+x^2-3x+5).$$






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              2 Answers
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              2 Answers
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              $begingroup$

              Since you know that $ lambda = 1$ works, note that your equation factors as
              $$(lambda - 1) (x^2 - 3x + 5 + lambda^2) = 0.$$
              This gives that
              $$lambda = 1, lambda = pm sqrt{3x - x^2 - 5}.$$






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                Since you know that $ lambda = 1$ works, note that your equation factors as
                $$(lambda - 1) (x^2 - 3x + 5 + lambda^2) = 0.$$
                This gives that
                $$lambda = 1, lambda = pm sqrt{3x - x^2 - 5}.$$






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  Since you know that $ lambda = 1$ works, note that your equation factors as
                  $$(lambda - 1) (x^2 - 3x + 5 + lambda^2) = 0.$$
                  This gives that
                  $$lambda = 1, lambda = pm sqrt{3x - x^2 - 5}.$$






                  share|cite|improve this answer









                  $endgroup$



                  Since you know that $ lambda = 1$ works, note that your equation factors as
                  $$(lambda - 1) (x^2 - 3x + 5 + lambda^2) = 0.$$
                  This gives that
                  $$lambda = 1, lambda = pm sqrt{3x - x^2 - 5}.$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Jan 5 at 22:58









                  Klint QinamiKlint Qinami

                  1,137510




                  1,137510























                      1












                      $begingroup$

                      Since you know that $1$ is a root of your polynomial, divide it by $lambda-1$, getting$$lambda^3-lambda^2-2xlambda+5lambda+x^2lambda-xlambda+3x-x^2-5=(lambda-1)(lambda^2+x^2-3x+5).$$






                      share|cite|improve this answer









                      $endgroup$


















                        1












                        $begingroup$

                        Since you know that $1$ is a root of your polynomial, divide it by $lambda-1$, getting$$lambda^3-lambda^2-2xlambda+5lambda+x^2lambda-xlambda+3x-x^2-5=(lambda-1)(lambda^2+x^2-3x+5).$$






                        share|cite|improve this answer









                        $endgroup$
















                          1












                          1








                          1





                          $begingroup$

                          Since you know that $1$ is a root of your polynomial, divide it by $lambda-1$, getting$$lambda^3-lambda^2-2xlambda+5lambda+x^2lambda-xlambda+3x-x^2-5=(lambda-1)(lambda^2+x^2-3x+5).$$






                          share|cite|improve this answer









                          $endgroup$



                          Since you know that $1$ is a root of your polynomial, divide it by $lambda-1$, getting$$lambda^3-lambda^2-2xlambda+5lambda+x^2lambda-xlambda+3x-x^2-5=(lambda-1)(lambda^2+x^2-3x+5).$$







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Jan 5 at 22:59









                          José Carlos SantosJosé Carlos Santos

                          173k23133241




                          173k23133241






























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