$4 times 4$ matrix and homogeneous system of equations.












0












$begingroup$


I have the following question here:




Let $A$ be a $4 times 4$ matrix such that $x=begin{matrix}[-4 0 2 -8]^T end{matrix}$ is a solution to the homogeneous system of linear equations $Ax=0$. Which of the following statements is false?



$(a)$ $det(A)=0$.



$(b)$ The linear system $Ax=b$ is consistent for every $ 4 times 1$ column vector b.



$(c)$ The reduced row echelon form of $A$ has at least one row of zeroes.



$(d)$ $x=begin{matrix}[6 0 -3 12]end{matrix}$ is also a solution to $Ax=0$



$(e)$ There exists a positive integer r so that the linear system $Ax=0$ has $4-$r free variables.




The answer is supposed to be $(b)$ but I'm just not seeing why. I thought that for a homogeneous system of equations, the solutions were always unique or there were infinitely many solutions.



Why is $(b)$ false? Does the statement not imply that we have a consistent system of equations for any b since it is a homogeneous system?



Also:



Why is $(a)$ true? What does the determinant have to do with this?



For $(c)$, how do we know the matrix has a row of zeroes? That would normally mean we have a free variable but I don't see how that's possible here? Why can we assume there are infinitely many solutions?



Why is $(d)$ true? I thought if the system is consistent, there is only one such solutions. How can $x$ have two sets of solutions?



For $(e)$ is there some sort of theorem I am missing?



My theory for linear algebra is fairly weak as you might think... I am decent at it but I can't just wrap my head around this.










share|cite|improve this question











$endgroup$












  • $begingroup$
    You have not mentioned the option ($e$) in your question.
    $endgroup$
    – Yadati Kiran
    Dec 18 '18 at 7:07










  • $begingroup$
    Thanks. I edited.
    $endgroup$
    – Future Math person
    Dec 18 '18 at 7:15






  • 1




    $begingroup$
    $Ax=b$ is not homogeneous.
    $endgroup$
    – amd
    Dec 18 '18 at 7:49
















0












$begingroup$


I have the following question here:




Let $A$ be a $4 times 4$ matrix such that $x=begin{matrix}[-4 0 2 -8]^T end{matrix}$ is a solution to the homogeneous system of linear equations $Ax=0$. Which of the following statements is false?



$(a)$ $det(A)=0$.



$(b)$ The linear system $Ax=b$ is consistent for every $ 4 times 1$ column vector b.



$(c)$ The reduced row echelon form of $A$ has at least one row of zeroes.



$(d)$ $x=begin{matrix}[6 0 -3 12]end{matrix}$ is also a solution to $Ax=0$



$(e)$ There exists a positive integer r so that the linear system $Ax=0$ has $4-$r free variables.




The answer is supposed to be $(b)$ but I'm just not seeing why. I thought that for a homogeneous system of equations, the solutions were always unique or there were infinitely many solutions.



Why is $(b)$ false? Does the statement not imply that we have a consistent system of equations for any b since it is a homogeneous system?



Also:



Why is $(a)$ true? What does the determinant have to do with this?



For $(c)$, how do we know the matrix has a row of zeroes? That would normally mean we have a free variable but I don't see how that's possible here? Why can we assume there are infinitely many solutions?



Why is $(d)$ true? I thought if the system is consistent, there is only one such solutions. How can $x$ have two sets of solutions?



For $(e)$ is there some sort of theorem I am missing?



My theory for linear algebra is fairly weak as you might think... I am decent at it but I can't just wrap my head around this.










share|cite|improve this question











$endgroup$












  • $begingroup$
    You have not mentioned the option ($e$) in your question.
    $endgroup$
    – Yadati Kiran
    Dec 18 '18 at 7:07










  • $begingroup$
    Thanks. I edited.
    $endgroup$
    – Future Math person
    Dec 18 '18 at 7:15






  • 1




    $begingroup$
    $Ax=b$ is not homogeneous.
    $endgroup$
    – amd
    Dec 18 '18 at 7:49














0












0








0





$begingroup$


I have the following question here:




Let $A$ be a $4 times 4$ matrix such that $x=begin{matrix}[-4 0 2 -8]^T end{matrix}$ is a solution to the homogeneous system of linear equations $Ax=0$. Which of the following statements is false?



$(a)$ $det(A)=0$.



$(b)$ The linear system $Ax=b$ is consistent for every $ 4 times 1$ column vector b.



$(c)$ The reduced row echelon form of $A$ has at least one row of zeroes.



$(d)$ $x=begin{matrix}[6 0 -3 12]end{matrix}$ is also a solution to $Ax=0$



$(e)$ There exists a positive integer r so that the linear system $Ax=0$ has $4-$r free variables.




The answer is supposed to be $(b)$ but I'm just not seeing why. I thought that for a homogeneous system of equations, the solutions were always unique or there were infinitely many solutions.



Why is $(b)$ false? Does the statement not imply that we have a consistent system of equations for any b since it is a homogeneous system?



Also:



Why is $(a)$ true? What does the determinant have to do with this?



For $(c)$, how do we know the matrix has a row of zeroes? That would normally mean we have a free variable but I don't see how that's possible here? Why can we assume there are infinitely many solutions?



Why is $(d)$ true? I thought if the system is consistent, there is only one such solutions. How can $x$ have two sets of solutions?



For $(e)$ is there some sort of theorem I am missing?



My theory for linear algebra is fairly weak as you might think... I am decent at it but I can't just wrap my head around this.










share|cite|improve this question











$endgroup$




I have the following question here:




Let $A$ be a $4 times 4$ matrix such that $x=begin{matrix}[-4 0 2 -8]^T end{matrix}$ is a solution to the homogeneous system of linear equations $Ax=0$. Which of the following statements is false?



$(a)$ $det(A)=0$.



$(b)$ The linear system $Ax=b$ is consistent for every $ 4 times 1$ column vector b.



$(c)$ The reduced row echelon form of $A$ has at least one row of zeroes.



$(d)$ $x=begin{matrix}[6 0 -3 12]end{matrix}$ is also a solution to $Ax=0$



$(e)$ There exists a positive integer r so that the linear system $Ax=0$ has $4-$r free variables.




The answer is supposed to be $(b)$ but I'm just not seeing why. I thought that for a homogeneous system of equations, the solutions were always unique or there were infinitely many solutions.



Why is $(b)$ false? Does the statement not imply that we have a consistent system of equations for any b since it is a homogeneous system?



Also:



Why is $(a)$ true? What does the determinant have to do with this?



For $(c)$, how do we know the matrix has a row of zeroes? That would normally mean we have a free variable but I don't see how that's possible here? Why can we assume there are infinitely many solutions?



Why is $(d)$ true? I thought if the system is consistent, there is only one such solutions. How can $x$ have two sets of solutions?



For $(e)$ is there some sort of theorem I am missing?



My theory for linear algebra is fairly weak as you might think... I am decent at it but I can't just wrap my head around this.







linear-algebra systems-of-equations determinant






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share|cite|improve this question













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share|cite|improve this question








edited Dec 18 '18 at 7:51







Future Math person

















asked Dec 18 '18 at 7:02









Future Math personFuture Math person

972817




972817












  • $begingroup$
    You have not mentioned the option ($e$) in your question.
    $endgroup$
    – Yadati Kiran
    Dec 18 '18 at 7:07










  • $begingroup$
    Thanks. I edited.
    $endgroup$
    – Future Math person
    Dec 18 '18 at 7:15






  • 1




    $begingroup$
    $Ax=b$ is not homogeneous.
    $endgroup$
    – amd
    Dec 18 '18 at 7:49


















  • $begingroup$
    You have not mentioned the option ($e$) in your question.
    $endgroup$
    – Yadati Kiran
    Dec 18 '18 at 7:07










  • $begingroup$
    Thanks. I edited.
    $endgroup$
    – Future Math person
    Dec 18 '18 at 7:15






  • 1




    $begingroup$
    $Ax=b$ is not homogeneous.
    $endgroup$
    – amd
    Dec 18 '18 at 7:49
















$begingroup$
You have not mentioned the option ($e$) in your question.
$endgroup$
– Yadati Kiran
Dec 18 '18 at 7:07




$begingroup$
You have not mentioned the option ($e$) in your question.
$endgroup$
– Yadati Kiran
Dec 18 '18 at 7:07












$begingroup$
Thanks. I edited.
$endgroup$
– Future Math person
Dec 18 '18 at 7:15




$begingroup$
Thanks. I edited.
$endgroup$
– Future Math person
Dec 18 '18 at 7:15




1




1




$begingroup$
$Ax=b$ is not homogeneous.
$endgroup$
– amd
Dec 18 '18 at 7:49




$begingroup$
$Ax=b$ is not homogeneous.
$endgroup$
– amd
Dec 18 '18 at 7:49










3 Answers
3






active

oldest

votes


















3












$begingroup$

One of the many equivalent conditions for a matrix $A$ to be invertible is that the only solution to $Ax=0$ is the trivial solution i.e. if $Ax=0$ then $x=0$. In this problem, you are given a nontrivial solution to $Ax=0$ so we know $A$ must not be invertible.



This immediately implies that $detA = 0$, so $(a)$ is true. Statements $(c)$ and $(e)$ are essentially saying the same thing: each row of zeroes in the reduced row echelon form of $A$ will correspond to a free variable in the equation $Ax=0$.



This leaves $(b)$. "I thought that for a homogeneous system of equations, the solutions were always unique or there were infinitely many solutions." Yes, this is true if there are solutions. However, there will be some vectors $b$ such that $Ax=b$ does not have a solution. The reason for that is because $A$ will have at most $3$ pivots, not the maximum $4$. Since the columns of $A$ are not linearly independent, there will be vectors $b$ that cannot be expressed as a linear combination of the columns. Hence $(b)$ is false.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Hey so I edited my question and added the correct choice $(d)$. In your edit, it should say "Statements $(c)$ and $(e)$ are essentially saying the same thing: each...." I get everything you said now but why is choice $(d)$ (The one I just put in) true? I thought that if a system of equations has one solutions, it can't have another.
    $endgroup$
    – Future Math person
    Dec 18 '18 at 7:44












  • $begingroup$
    Never mind! It's because any scalar multiple of the solutions is also a valid solution.
    $endgroup$
    – Future Math person
    Dec 18 '18 at 7:56










  • $begingroup$
    $(e)$ is actually untrue for $A=O$
    $endgroup$
    – Shubham Johri
    Dec 18 '18 at 8:06










  • $begingroup$
    @FutureMathperson Exactly!
    $endgroup$
    – pwerth
    Dec 18 '18 at 16:02










  • $begingroup$
    @ShubhamJohri Fair point, this probably should have been excluded in the question
    $endgroup$
    – pwerth
    Dec 18 '18 at 16:03



















4












$begingroup$

The reason that (b) is false is because the only way for the system to be consistent for any column vector $b$ would be if the four columns of $A$ were linearly independent, which in turn can only be the case if $Ax = 0$ admits only the solution $x = 0$.






share|cite|improve this answer









$endgroup$





















    0












    $begingroup$

    (b) is false because $Ax=[-4;0;2;-8]^T$ is not consistent.






    share|cite|improve this answer









    $endgroup$













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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      3












      $begingroup$

      One of the many equivalent conditions for a matrix $A$ to be invertible is that the only solution to $Ax=0$ is the trivial solution i.e. if $Ax=0$ then $x=0$. In this problem, you are given a nontrivial solution to $Ax=0$ so we know $A$ must not be invertible.



      This immediately implies that $detA = 0$, so $(a)$ is true. Statements $(c)$ and $(e)$ are essentially saying the same thing: each row of zeroes in the reduced row echelon form of $A$ will correspond to a free variable in the equation $Ax=0$.



      This leaves $(b)$. "I thought that for a homogeneous system of equations, the solutions were always unique or there were infinitely many solutions." Yes, this is true if there are solutions. However, there will be some vectors $b$ such that $Ax=b$ does not have a solution. The reason for that is because $A$ will have at most $3$ pivots, not the maximum $4$. Since the columns of $A$ are not linearly independent, there will be vectors $b$ that cannot be expressed as a linear combination of the columns. Hence $(b)$ is false.






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        Hey so I edited my question and added the correct choice $(d)$. In your edit, it should say "Statements $(c)$ and $(e)$ are essentially saying the same thing: each...." I get everything you said now but why is choice $(d)$ (The one I just put in) true? I thought that if a system of equations has one solutions, it can't have another.
        $endgroup$
        – Future Math person
        Dec 18 '18 at 7:44












      • $begingroup$
        Never mind! It's because any scalar multiple of the solutions is also a valid solution.
        $endgroup$
        – Future Math person
        Dec 18 '18 at 7:56










      • $begingroup$
        $(e)$ is actually untrue for $A=O$
        $endgroup$
        – Shubham Johri
        Dec 18 '18 at 8:06










      • $begingroup$
        @FutureMathperson Exactly!
        $endgroup$
        – pwerth
        Dec 18 '18 at 16:02










      • $begingroup$
        @ShubhamJohri Fair point, this probably should have been excluded in the question
        $endgroup$
        – pwerth
        Dec 18 '18 at 16:03
















      3












      $begingroup$

      One of the many equivalent conditions for a matrix $A$ to be invertible is that the only solution to $Ax=0$ is the trivial solution i.e. if $Ax=0$ then $x=0$. In this problem, you are given a nontrivial solution to $Ax=0$ so we know $A$ must not be invertible.



      This immediately implies that $detA = 0$, so $(a)$ is true. Statements $(c)$ and $(e)$ are essentially saying the same thing: each row of zeroes in the reduced row echelon form of $A$ will correspond to a free variable in the equation $Ax=0$.



      This leaves $(b)$. "I thought that for a homogeneous system of equations, the solutions were always unique or there were infinitely many solutions." Yes, this is true if there are solutions. However, there will be some vectors $b$ such that $Ax=b$ does not have a solution. The reason for that is because $A$ will have at most $3$ pivots, not the maximum $4$. Since the columns of $A$ are not linearly independent, there will be vectors $b$ that cannot be expressed as a linear combination of the columns. Hence $(b)$ is false.






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        Hey so I edited my question and added the correct choice $(d)$. In your edit, it should say "Statements $(c)$ and $(e)$ are essentially saying the same thing: each...." I get everything you said now but why is choice $(d)$ (The one I just put in) true? I thought that if a system of equations has one solutions, it can't have another.
        $endgroup$
        – Future Math person
        Dec 18 '18 at 7:44












      • $begingroup$
        Never mind! It's because any scalar multiple of the solutions is also a valid solution.
        $endgroup$
        – Future Math person
        Dec 18 '18 at 7:56










      • $begingroup$
        $(e)$ is actually untrue for $A=O$
        $endgroup$
        – Shubham Johri
        Dec 18 '18 at 8:06










      • $begingroup$
        @FutureMathperson Exactly!
        $endgroup$
        – pwerth
        Dec 18 '18 at 16:02










      • $begingroup$
        @ShubhamJohri Fair point, this probably should have been excluded in the question
        $endgroup$
        – pwerth
        Dec 18 '18 at 16:03














      3












      3








      3





      $begingroup$

      One of the many equivalent conditions for a matrix $A$ to be invertible is that the only solution to $Ax=0$ is the trivial solution i.e. if $Ax=0$ then $x=0$. In this problem, you are given a nontrivial solution to $Ax=0$ so we know $A$ must not be invertible.



      This immediately implies that $detA = 0$, so $(a)$ is true. Statements $(c)$ and $(e)$ are essentially saying the same thing: each row of zeroes in the reduced row echelon form of $A$ will correspond to a free variable in the equation $Ax=0$.



      This leaves $(b)$. "I thought that for a homogeneous system of equations, the solutions were always unique or there were infinitely many solutions." Yes, this is true if there are solutions. However, there will be some vectors $b$ such that $Ax=b$ does not have a solution. The reason for that is because $A$ will have at most $3$ pivots, not the maximum $4$. Since the columns of $A$ are not linearly independent, there will be vectors $b$ that cannot be expressed as a linear combination of the columns. Hence $(b)$ is false.






      share|cite|improve this answer











      $endgroup$



      One of the many equivalent conditions for a matrix $A$ to be invertible is that the only solution to $Ax=0$ is the trivial solution i.e. if $Ax=0$ then $x=0$. In this problem, you are given a nontrivial solution to $Ax=0$ so we know $A$ must not be invertible.



      This immediately implies that $detA = 0$, so $(a)$ is true. Statements $(c)$ and $(e)$ are essentially saying the same thing: each row of zeroes in the reduced row echelon form of $A$ will correspond to a free variable in the equation $Ax=0$.



      This leaves $(b)$. "I thought that for a homogeneous system of equations, the solutions were always unique or there were infinitely many solutions." Yes, this is true if there are solutions. However, there will be some vectors $b$ such that $Ax=b$ does not have a solution. The reason for that is because $A$ will have at most $3$ pivots, not the maximum $4$. Since the columns of $A$ are not linearly independent, there will be vectors $b$ that cannot be expressed as a linear combination of the columns. Hence $(b)$ is false.







      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited Dec 18 '18 at 16:02

























      answered Dec 18 '18 at 7:10









      pwerthpwerth

      3,243417




      3,243417












      • $begingroup$
        Hey so I edited my question and added the correct choice $(d)$. In your edit, it should say "Statements $(c)$ and $(e)$ are essentially saying the same thing: each...." I get everything you said now but why is choice $(d)$ (The one I just put in) true? I thought that if a system of equations has one solutions, it can't have another.
        $endgroup$
        – Future Math person
        Dec 18 '18 at 7:44












      • $begingroup$
        Never mind! It's because any scalar multiple of the solutions is also a valid solution.
        $endgroup$
        – Future Math person
        Dec 18 '18 at 7:56










      • $begingroup$
        $(e)$ is actually untrue for $A=O$
        $endgroup$
        – Shubham Johri
        Dec 18 '18 at 8:06










      • $begingroup$
        @FutureMathperson Exactly!
        $endgroup$
        – pwerth
        Dec 18 '18 at 16:02










      • $begingroup$
        @ShubhamJohri Fair point, this probably should have been excluded in the question
        $endgroup$
        – pwerth
        Dec 18 '18 at 16:03


















      • $begingroup$
        Hey so I edited my question and added the correct choice $(d)$. In your edit, it should say "Statements $(c)$ and $(e)$ are essentially saying the same thing: each...." I get everything you said now but why is choice $(d)$ (The one I just put in) true? I thought that if a system of equations has one solutions, it can't have another.
        $endgroup$
        – Future Math person
        Dec 18 '18 at 7:44












      • $begingroup$
        Never mind! It's because any scalar multiple of the solutions is also a valid solution.
        $endgroup$
        – Future Math person
        Dec 18 '18 at 7:56










      • $begingroup$
        $(e)$ is actually untrue for $A=O$
        $endgroup$
        – Shubham Johri
        Dec 18 '18 at 8:06










      • $begingroup$
        @FutureMathperson Exactly!
        $endgroup$
        – pwerth
        Dec 18 '18 at 16:02










      • $begingroup$
        @ShubhamJohri Fair point, this probably should have been excluded in the question
        $endgroup$
        – pwerth
        Dec 18 '18 at 16:03
















      $begingroup$
      Hey so I edited my question and added the correct choice $(d)$. In your edit, it should say "Statements $(c)$ and $(e)$ are essentially saying the same thing: each...." I get everything you said now but why is choice $(d)$ (The one I just put in) true? I thought that if a system of equations has one solutions, it can't have another.
      $endgroup$
      – Future Math person
      Dec 18 '18 at 7:44






      $begingroup$
      Hey so I edited my question and added the correct choice $(d)$. In your edit, it should say "Statements $(c)$ and $(e)$ are essentially saying the same thing: each...." I get everything you said now but why is choice $(d)$ (The one I just put in) true? I thought that if a system of equations has one solutions, it can't have another.
      $endgroup$
      – Future Math person
      Dec 18 '18 at 7:44














      $begingroup$
      Never mind! It's because any scalar multiple of the solutions is also a valid solution.
      $endgroup$
      – Future Math person
      Dec 18 '18 at 7:56




      $begingroup$
      Never mind! It's because any scalar multiple of the solutions is also a valid solution.
      $endgroup$
      – Future Math person
      Dec 18 '18 at 7:56












      $begingroup$
      $(e)$ is actually untrue for $A=O$
      $endgroup$
      – Shubham Johri
      Dec 18 '18 at 8:06




      $begingroup$
      $(e)$ is actually untrue for $A=O$
      $endgroup$
      – Shubham Johri
      Dec 18 '18 at 8:06












      $begingroup$
      @FutureMathperson Exactly!
      $endgroup$
      – pwerth
      Dec 18 '18 at 16:02




      $begingroup$
      @FutureMathperson Exactly!
      $endgroup$
      – pwerth
      Dec 18 '18 at 16:02












      $begingroup$
      @ShubhamJohri Fair point, this probably should have been excluded in the question
      $endgroup$
      – pwerth
      Dec 18 '18 at 16:03




      $begingroup$
      @ShubhamJohri Fair point, this probably should have been excluded in the question
      $endgroup$
      – pwerth
      Dec 18 '18 at 16:03











      4












      $begingroup$

      The reason that (b) is false is because the only way for the system to be consistent for any column vector $b$ would be if the four columns of $A$ were linearly independent, which in turn can only be the case if $Ax = 0$ admits only the solution $x = 0$.






      share|cite|improve this answer









      $endgroup$


















        4












        $begingroup$

        The reason that (b) is false is because the only way for the system to be consistent for any column vector $b$ would be if the four columns of $A$ were linearly independent, which in turn can only be the case if $Ax = 0$ admits only the solution $x = 0$.






        share|cite|improve this answer









        $endgroup$
















          4












          4








          4





          $begingroup$

          The reason that (b) is false is because the only way for the system to be consistent for any column vector $b$ would be if the four columns of $A$ were linearly independent, which in turn can only be the case if $Ax = 0$ admits only the solution $x = 0$.






          share|cite|improve this answer









          $endgroup$



          The reason that (b) is false is because the only way for the system to be consistent for any column vector $b$ would be if the four columns of $A$ were linearly independent, which in turn can only be the case if $Ax = 0$ admits only the solution $x = 0$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 18 '18 at 7:09









          D.B.D.B.

          1,24318




          1,24318























              0












              $begingroup$

              (b) is false because $Ax=[-4;0;2;-8]^T$ is not consistent.






              share|cite|improve this answer









              $endgroup$


















                0












                $begingroup$

                (b) is false because $Ax=[-4;0;2;-8]^T$ is not consistent.






                share|cite|improve this answer









                $endgroup$
















                  0












                  0








                  0





                  $begingroup$

                  (b) is false because $Ax=[-4;0;2;-8]^T$ is not consistent.






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                  $endgroup$



                  (b) is false because $Ax=[-4;0;2;-8]^T$ is not consistent.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 18 '18 at 7:14









                  A. GomezA. Gomez

                  256




                  256






























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