Need help solving for a linear system $x_1$ + $2x_2$ = $λx_1$ & $2x_1$ + $2x_2$ = $λx_2$












0














I'm working on an exercise problem that is as follows. Find all real values of λ for which the system has a non trivial solution



$x_1$ + $2x_2$ = $λx_1$



$2x_1$ + $x_2$ = $λx_2$



I'm not sure if I did it correctly but I rewrote the equation as:



$$
begin{matrix}
1 & 2\
2 & 1\
end{matrix}
$$



multiplied by



$$
begin{matrix}
x_1\
x_2\
end{matrix}
$$



which equals



$$
begin{matrix}
λx_1\
λx_2\
end{matrix}
$$



I then multiplied the both sides by the inverse of the the 2x2 matrix giving me just
$$
begin{matrix}
x_1\
x_2\
end{matrix}
$$



on the left hand side and I solved for $x_1$ and $x_2$ in terms of $λx_1$ and $λx_2$. After this I plugged the values I got for $x_1$ and $x_2$ back into the first equation in the linear system. I feel like I'm on the wrong track, so any help on how to solve this would be appreciated.










share|cite|improve this question
























  • Sorry I corrected it. I wrote down the intial linear system wrong.
    – PCR
    Feb 17 '16 at 20:32










  • Move everything to one side of the equal sign, correct like terms, turn it into a matrix. Solve for when the determinant is zero.
    – Kaynex
    Feb 17 '16 at 20:51
















0














I'm working on an exercise problem that is as follows. Find all real values of λ for which the system has a non trivial solution



$x_1$ + $2x_2$ = $λx_1$



$2x_1$ + $x_2$ = $λx_2$



I'm not sure if I did it correctly but I rewrote the equation as:



$$
begin{matrix}
1 & 2\
2 & 1\
end{matrix}
$$



multiplied by



$$
begin{matrix}
x_1\
x_2\
end{matrix}
$$



which equals



$$
begin{matrix}
λx_1\
λx_2\
end{matrix}
$$



I then multiplied the both sides by the inverse of the the 2x2 matrix giving me just
$$
begin{matrix}
x_1\
x_2\
end{matrix}
$$



on the left hand side and I solved for $x_1$ and $x_2$ in terms of $λx_1$ and $λx_2$. After this I plugged the values I got for $x_1$ and $x_2$ back into the first equation in the linear system. I feel like I'm on the wrong track, so any help on how to solve this would be appreciated.










share|cite|improve this question
























  • Sorry I corrected it. I wrote down the intial linear system wrong.
    – PCR
    Feb 17 '16 at 20:32










  • Move everything to one side of the equal sign, correct like terms, turn it into a matrix. Solve for when the determinant is zero.
    – Kaynex
    Feb 17 '16 at 20:51














0












0








0


0





I'm working on an exercise problem that is as follows. Find all real values of λ for which the system has a non trivial solution



$x_1$ + $2x_2$ = $λx_1$



$2x_1$ + $x_2$ = $λx_2$



I'm not sure if I did it correctly but I rewrote the equation as:



$$
begin{matrix}
1 & 2\
2 & 1\
end{matrix}
$$



multiplied by



$$
begin{matrix}
x_1\
x_2\
end{matrix}
$$



which equals



$$
begin{matrix}
λx_1\
λx_2\
end{matrix}
$$



I then multiplied the both sides by the inverse of the the 2x2 matrix giving me just
$$
begin{matrix}
x_1\
x_2\
end{matrix}
$$



on the left hand side and I solved for $x_1$ and $x_2$ in terms of $λx_1$ and $λx_2$. After this I plugged the values I got for $x_1$ and $x_2$ back into the first equation in the linear system. I feel like I'm on the wrong track, so any help on how to solve this would be appreciated.










share|cite|improve this question















I'm working on an exercise problem that is as follows. Find all real values of λ for which the system has a non trivial solution



$x_1$ + $2x_2$ = $λx_1$



$2x_1$ + $x_2$ = $λx_2$



I'm not sure if I did it correctly but I rewrote the equation as:



$$
begin{matrix}
1 & 2\
2 & 1\
end{matrix}
$$



multiplied by



$$
begin{matrix}
x_1\
x_2\
end{matrix}
$$



which equals



$$
begin{matrix}
λx_1\
λx_2\
end{matrix}
$$



I then multiplied the both sides by the inverse of the the 2x2 matrix giving me just
$$
begin{matrix}
x_1\
x_2\
end{matrix}
$$



on the left hand side and I solved for $x_1$ and $x_2$ in terms of $λx_1$ and $λx_2$. After this I plugged the values I got for $x_1$ and $x_2$ back into the first equation in the linear system. I feel like I'm on the wrong track, so any help on how to solve this would be appreciated.







linear-algebra






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 5 '18 at 8:31









Math Girl

635318




635318










asked Feb 17 '16 at 20:23









PCRPCR

216




216












  • Sorry I corrected it. I wrote down the intial linear system wrong.
    – PCR
    Feb 17 '16 at 20:32










  • Move everything to one side of the equal sign, correct like terms, turn it into a matrix. Solve for when the determinant is zero.
    – Kaynex
    Feb 17 '16 at 20:51


















  • Sorry I corrected it. I wrote down the intial linear system wrong.
    – PCR
    Feb 17 '16 at 20:32










  • Move everything to one side of the equal sign, correct like terms, turn it into a matrix. Solve for when the determinant is zero.
    – Kaynex
    Feb 17 '16 at 20:51
















Sorry I corrected it. I wrote down the intial linear system wrong.
– PCR
Feb 17 '16 at 20:32




Sorry I corrected it. I wrote down the intial linear system wrong.
– PCR
Feb 17 '16 at 20:32












Move everything to one side of the equal sign, correct like terms, turn it into a matrix. Solve for when the determinant is zero.
– Kaynex
Feb 17 '16 at 20:51




Move everything to one side of the equal sign, correct like terms, turn it into a matrix. Solve for when the determinant is zero.
– Kaynex
Feb 17 '16 at 20:51










3 Answers
3






active

oldest

votes


















1














If you do not know linear algebra, then rewrite the first equation to $x_2=frac{x_1(lambda-1)}{2}$ and substitute this into the second equation. Then we obtain
$$
(lambda+1)(lambda-3)x_1=0.
$$
Now argue that we have a non-trivial solution only for $(lambda+1)(lambda-3)=0$.






share|cite|improve this answer





























    0














    You have



    $$begin{bmatrix}
    1 & 2 \
    2 & 2
    end{bmatrix} begin{bmatrix}
    x_1 \
    x_2
    end{bmatrix}=
    lambda begin{bmatrix}
    x_1 \
    x_2
    end{bmatrix},$$
    so $lambda$ is eigenvalue of matrix $begin{bmatrix}
    1 & 2 \
    2 & 2
    end{bmatrix}$. Can you find all eigenvalues of that matrix using, for example, characteristic polynomial?






    share|cite|improve this answer





















    • I don't know what an eigenvalue is. Looking ahead in the textbook, it's a topic in the next section, so I'm assuming the textbook wants me to solve it without knowing what an eigenvalue is.
      – PCR
      Feb 17 '16 at 20:34










    • @PCR what if your textbook wants to lure you to the idea of eigenvctors and eigenvalues? In my time, I had much, much more trouble with those teachers who did not want me to look ahead.
      – Gyro Gearloose
      Feb 17 '16 at 20:42



















    0














    The mistake you made: You have chosen $A=begin{bmatrix}1 & 2\2 & 1end{bmatrix}$ and $x=begin{bmatrix}x_1\x_2end{bmatrix}$. So you have $Ax=lambda x$ in the matrix equation form. When you multiply $A^{-1}$, you will get $x=Ix=A^{-1}Ax=lambda A^{-1}x$. Observe the right hand side involves another matrix $A^{-1}$. Hence you are not getting the solution. Consider the technique suggested by 'Dietrich Burde' and try to generalize. Your text book is probably designed in such a way that you can understand the generalizations in the next sections.






    share|cite|improve this answer





















      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      autoActivateHeartbeat: false,
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1660448%2fneed-help-solving-for-a-linear-system-x-1-2x-2-%25ce%25bbx-1-2x-1-2x-2%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      1














      If you do not know linear algebra, then rewrite the first equation to $x_2=frac{x_1(lambda-1)}{2}$ and substitute this into the second equation. Then we obtain
      $$
      (lambda+1)(lambda-3)x_1=0.
      $$
      Now argue that we have a non-trivial solution only for $(lambda+1)(lambda-3)=0$.






      share|cite|improve this answer


























        1














        If you do not know linear algebra, then rewrite the first equation to $x_2=frac{x_1(lambda-1)}{2}$ and substitute this into the second equation. Then we obtain
        $$
        (lambda+1)(lambda-3)x_1=0.
        $$
        Now argue that we have a non-trivial solution only for $(lambda+1)(lambda-3)=0$.






        share|cite|improve this answer
























          1












          1








          1






          If you do not know linear algebra, then rewrite the first equation to $x_2=frac{x_1(lambda-1)}{2}$ and substitute this into the second equation. Then we obtain
          $$
          (lambda+1)(lambda-3)x_1=0.
          $$
          Now argue that we have a non-trivial solution only for $(lambda+1)(lambda-3)=0$.






          share|cite|improve this answer












          If you do not know linear algebra, then rewrite the first equation to $x_2=frac{x_1(lambda-1)}{2}$ and substitute this into the second equation. Then we obtain
          $$
          (lambda+1)(lambda-3)x_1=0.
          $$
          Now argue that we have a non-trivial solution only for $(lambda+1)(lambda-3)=0$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Feb 17 '16 at 20:42









          Dietrich BurdeDietrich Burde

          78.1k64386




          78.1k64386























              0














              You have



              $$begin{bmatrix}
              1 & 2 \
              2 & 2
              end{bmatrix} begin{bmatrix}
              x_1 \
              x_2
              end{bmatrix}=
              lambda begin{bmatrix}
              x_1 \
              x_2
              end{bmatrix},$$
              so $lambda$ is eigenvalue of matrix $begin{bmatrix}
              1 & 2 \
              2 & 2
              end{bmatrix}$. Can you find all eigenvalues of that matrix using, for example, characteristic polynomial?






              share|cite|improve this answer





















              • I don't know what an eigenvalue is. Looking ahead in the textbook, it's a topic in the next section, so I'm assuming the textbook wants me to solve it without knowing what an eigenvalue is.
                – PCR
                Feb 17 '16 at 20:34










              • @PCR what if your textbook wants to lure you to the idea of eigenvctors and eigenvalues? In my time, I had much, much more trouble with those teachers who did not want me to look ahead.
                – Gyro Gearloose
                Feb 17 '16 at 20:42
















              0














              You have



              $$begin{bmatrix}
              1 & 2 \
              2 & 2
              end{bmatrix} begin{bmatrix}
              x_1 \
              x_2
              end{bmatrix}=
              lambda begin{bmatrix}
              x_1 \
              x_2
              end{bmatrix},$$
              so $lambda$ is eigenvalue of matrix $begin{bmatrix}
              1 & 2 \
              2 & 2
              end{bmatrix}$. Can you find all eigenvalues of that matrix using, for example, characteristic polynomial?






              share|cite|improve this answer





















              • I don't know what an eigenvalue is. Looking ahead in the textbook, it's a topic in the next section, so I'm assuming the textbook wants me to solve it without knowing what an eigenvalue is.
                – PCR
                Feb 17 '16 at 20:34










              • @PCR what if your textbook wants to lure you to the idea of eigenvctors and eigenvalues? In my time, I had much, much more trouble with those teachers who did not want me to look ahead.
                – Gyro Gearloose
                Feb 17 '16 at 20:42














              0












              0








              0






              You have



              $$begin{bmatrix}
              1 & 2 \
              2 & 2
              end{bmatrix} begin{bmatrix}
              x_1 \
              x_2
              end{bmatrix}=
              lambda begin{bmatrix}
              x_1 \
              x_2
              end{bmatrix},$$
              so $lambda$ is eigenvalue of matrix $begin{bmatrix}
              1 & 2 \
              2 & 2
              end{bmatrix}$. Can you find all eigenvalues of that matrix using, for example, characteristic polynomial?






              share|cite|improve this answer












              You have



              $$begin{bmatrix}
              1 & 2 \
              2 & 2
              end{bmatrix} begin{bmatrix}
              x_1 \
              x_2
              end{bmatrix}=
              lambda begin{bmatrix}
              x_1 \
              x_2
              end{bmatrix},$$
              so $lambda$ is eigenvalue of matrix $begin{bmatrix}
              1 & 2 \
              2 & 2
              end{bmatrix}$. Can you find all eigenvalues of that matrix using, for example, characteristic polynomial?







              share|cite|improve this answer












              share|cite|improve this answer



              share|cite|improve this answer










              answered Feb 17 '16 at 20:30









              aghaagha

              8,99641533




              8,99641533












              • I don't know what an eigenvalue is. Looking ahead in the textbook, it's a topic in the next section, so I'm assuming the textbook wants me to solve it without knowing what an eigenvalue is.
                – PCR
                Feb 17 '16 at 20:34










              • @PCR what if your textbook wants to lure you to the idea of eigenvctors and eigenvalues? In my time, I had much, much more trouble with those teachers who did not want me to look ahead.
                – Gyro Gearloose
                Feb 17 '16 at 20:42


















              • I don't know what an eigenvalue is. Looking ahead in the textbook, it's a topic in the next section, so I'm assuming the textbook wants me to solve it without knowing what an eigenvalue is.
                – PCR
                Feb 17 '16 at 20:34










              • @PCR what if your textbook wants to lure you to the idea of eigenvctors and eigenvalues? In my time, I had much, much more trouble with those teachers who did not want me to look ahead.
                – Gyro Gearloose
                Feb 17 '16 at 20:42
















              I don't know what an eigenvalue is. Looking ahead in the textbook, it's a topic in the next section, so I'm assuming the textbook wants me to solve it without knowing what an eigenvalue is.
              – PCR
              Feb 17 '16 at 20:34




              I don't know what an eigenvalue is. Looking ahead in the textbook, it's a topic in the next section, so I'm assuming the textbook wants me to solve it without knowing what an eigenvalue is.
              – PCR
              Feb 17 '16 at 20:34












              @PCR what if your textbook wants to lure you to the idea of eigenvctors and eigenvalues? In my time, I had much, much more trouble with those teachers who did not want me to look ahead.
              – Gyro Gearloose
              Feb 17 '16 at 20:42




              @PCR what if your textbook wants to lure you to the idea of eigenvctors and eigenvalues? In my time, I had much, much more trouble with those teachers who did not want me to look ahead.
              – Gyro Gearloose
              Feb 17 '16 at 20:42











              0














              The mistake you made: You have chosen $A=begin{bmatrix}1 & 2\2 & 1end{bmatrix}$ and $x=begin{bmatrix}x_1\x_2end{bmatrix}$. So you have $Ax=lambda x$ in the matrix equation form. When you multiply $A^{-1}$, you will get $x=Ix=A^{-1}Ax=lambda A^{-1}x$. Observe the right hand side involves another matrix $A^{-1}$. Hence you are not getting the solution. Consider the technique suggested by 'Dietrich Burde' and try to generalize. Your text book is probably designed in such a way that you can understand the generalizations in the next sections.






              share|cite|improve this answer


























                0














                The mistake you made: You have chosen $A=begin{bmatrix}1 & 2\2 & 1end{bmatrix}$ and $x=begin{bmatrix}x_1\x_2end{bmatrix}$. So you have $Ax=lambda x$ in the matrix equation form. When you multiply $A^{-1}$, you will get $x=Ix=A^{-1}Ax=lambda A^{-1}x$. Observe the right hand side involves another matrix $A^{-1}$. Hence you are not getting the solution. Consider the technique suggested by 'Dietrich Burde' and try to generalize. Your text book is probably designed in such a way that you can understand the generalizations in the next sections.






                share|cite|improve this answer
























                  0












                  0








                  0






                  The mistake you made: You have chosen $A=begin{bmatrix}1 & 2\2 & 1end{bmatrix}$ and $x=begin{bmatrix}x_1\x_2end{bmatrix}$. So you have $Ax=lambda x$ in the matrix equation form. When you multiply $A^{-1}$, you will get $x=Ix=A^{-1}Ax=lambda A^{-1}x$. Observe the right hand side involves another matrix $A^{-1}$. Hence you are not getting the solution. Consider the technique suggested by 'Dietrich Burde' and try to generalize. Your text book is probably designed in such a way that you can understand the generalizations in the next sections.






                  share|cite|improve this answer












                  The mistake you made: You have chosen $A=begin{bmatrix}1 & 2\2 & 1end{bmatrix}$ and $x=begin{bmatrix}x_1\x_2end{bmatrix}$. So you have $Ax=lambda x$ in the matrix equation form. When you multiply $A^{-1}$, you will get $x=Ix=A^{-1}Ax=lambda A^{-1}x$. Observe the right hand side involves another matrix $A^{-1}$. Hence you are not getting the solution. Consider the technique suggested by 'Dietrich Burde' and try to generalize. Your text book is probably designed in such a way that you can understand the generalizations in the next sections.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Feb 18 '16 at 15:20









                  G_0_pi_i_eG_0_pi_i_e

                  603515




                  603515






























                      draft saved

                      draft discarded




















































                      Thanks for contributing an answer to Mathematics Stack Exchange!


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      Use MathJax to format equations. MathJax reference.


                      To learn more, see our tips on writing great answers.





                      Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                      Please pay close attention to the following guidance:


                      • Please be sure to answer the question. Provide details and share your research!

                      But avoid



                      • Asking for help, clarification, or responding to other answers.

                      • Making statements based on opinion; back them up with references or personal experience.


                      To learn more, see our tips on writing great answers.




                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1660448%2fneed-help-solving-for-a-linear-system-x-1-2x-2-%25ce%25bbx-1-2x-1-2x-2%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      Popular posts from this blog

                      Berounka

                      Sphinx de Gizeh

                      Different font size/position of beamer's navigation symbols template's content depending on regular/plain...