Results 1 to 7 of 7

Math Help - Find a 3x1 matrix x, with entries not all zero such that Ax = x

  1. #1
    Junior Member
    Joined
    Jul 2011
    Posts
    53

    Find a 3x1 matrix x, with entries not all zero such that Ax = x

    Given the matrix \begin{pmatrix} 1 & 2 & -1\\ 1 & 0 & 1 \\ 4 & -4 & 5 \end{pmatrix}, I've to find a vector x, such that Ax = x.

    I'm not quite sure what's the best way of solving a question like this, I can try some values that could be correct, doing some half educated guesses, but that just seems clumsy and slow. Any tips?
    Last edited by Lepzed; September 18th 2011 at 06:01 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: Find a 3x1 matrix x, with entries not all zero such that Ax = x

    If x=(x_1,x_2,x_3)^t then,

    Ax=x \Leftrightarrow \ldots \Leftrightarrow\begin{Bmatrix}-2x_2-x_3=0\\ x_1-x_2+x_3=0\\ 4x_1-4x_2+4x_3=0\end{matrix}

    Solving the linear system, we obtain all the solutions. For example, x=(3,1,-2) is a non zero solution.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jul 2011
    Posts
    53

    Re: Find a 3x1 matrix x, with entries not all zero such that Ax = x

    I think I still don't understand right, if Ax=x, then if x = (3, 1, -2), then:

    3\begin{pmatrix} 1 \\ 1 \\ 4 \end{pmatrix} + \begin{pmatrix} 2 \\ 0 \\ -4 \end{pmatrix}-2\begin{pmatrix} -1 \\ 1 \\ 5 \end{pmatrix} = \begin{pmatrix} 7 \\ 1 \\ -2 \end{pmatrix} \neq \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}

    What am I missing here? Why do you let the equations be 0? According to my textbook the answer (a answer?) should be \begin{pmatrix} -\frac{1}{2}r \\ \frac{1}{2}r \\ r \end{pmatrix}. I see how this is a solution:

    -\frac{1}{2}\begin{pmatrix} 1 \\ 1 \\ 4 \end{pmatrix} + \frac{1}{2}\begin{pmatrix} 2 \\ 0 \\ -4 \end{pmatrix}+1\begin{pmatrix} -1 \\ 1 \\ 5 \end{pmatrix} =\begin{pmatrix} -\frac{1}{2} \\ \frac{1}{2} \\ 1 \end{pmatrix}

    I know how to solve a system of equations, but I don't get this...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: Find a 3x1 matrix x, with entries not all zero such that Ax = x

    Quote Originally Posted by Lepzed View Post
    What am I missing here?
    a_{12}=-2 , and you have written 2 .
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Jul 2011
    Posts
    53

    Re: Find a 3x1 matrix x, with entries not all zero such that Ax = x

    Ah, that was a typo, it should indeed be '2', not '-2', my apologies for that. But still, how did you get to your answer? Why did you let the equations be 0?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: Find a 3x1 matrix x, with entries not all zero such that Ax = x

    With your reformulation



    and a non null solution is x=(-1,1,2)^t .
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Jul 2011
    Posts
    53

    Re: Find a 3x1 matrix x, with entries not all zero such that Ax = x

    Ah, so basically you subtracted an x_row from each row, to make the equation 0 and than you just fill in the blanks?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Please Help. A and Adj A with missing entries
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 24th 2010, 05:11 AM
  2. [SOLVED] How do you find matrix this kind of matrix?
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: February 4th 2010, 05:22 AM
  3. Matrices with integer entries
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 12th 2009, 07:54 AM
  4. Boxed Matrix entries
    Posted in the LaTeX Help Forum
    Replies: 2
    Last Post: November 28th 2009, 03:07 AM
  5. Matrix Entries Formula
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 17th 2008, 08:12 PM

Search Tags


/mathhelpforum @mathhelpforum