Search results

  1. C

    Find elements of a matrix such that its determinant is zero

    I just thought of something. Couldn't the coefficients in front of the ##x_1^2## and ##x_2^2## be zero? If so, then the determiannt expansion when set to zero would yield an equation for a line that passes through those 3 distinct points.
  2. C

    Find elements of a matrix such that its determinant is zero

    Yes, I thought it was too simple to describe it like that as I did not go about finding constants u,v,w. For the equation of the circle, it seems I still am not finding the constants, but seems a bit more informative than what I had formerly planned on doing.
  3. C

    Find elements of a matrix such that its determinant is zero

    Oh I see. Originally I thought the following would also satisfy the problem: ##[x_1 x_2]^T=u[a_1 a_2]+v[b_1 b_2]+w[c_1 c_2]## for some arbitrary constants u,v,w, such that not all of them are zero.
  4. C

    Find elements of a matrix such that its determinant is zero

    This is going to be a dumb question. After I find the coefficients, A, C, D, E, and F, I will have some equation that describes a circle. Does this equation by itself satisfy the requirements of the problem, where I am asked for ##[x_1; x_2]##? The equation for the circle will be an implicit...
  5. C

    Find elements of a matrix such that its determinant is zero

    Had to look up what a circumcircle is and the first link is pretty much my homework problem http://mathworld.wolfram.com/Circumcircle.html
  6. C

    Find elements of a matrix such that its determinant is zero

    So it is an equilateral triangle? That means the points lie at 120degrees apart from each other.
  7. C

    Find elements of a matrix such that its determinant is zero

    Yes got it! Just edited my previous post right as you posted that.
  8. C

    Find elements of a matrix such that its determinant is zero

    Oh got it, from looking at the unexpanded determinant (and later verified by looking at the expanded determinant), I see that the following coefficients for ##x_1^2## and ##x_2^2## For ##x_1^2##, there is ##-a_1b_1, a_1c_2, a_2b_1, -a2c1, -b_1c_2, b_2c_1## looks like just (# of points...
  9. C

    Find elements of a matrix such that its determinant is zero

    Okay, so the equation becomes: ##Ax_1^2+Cx_2^2+Dx_1+Ex_2+F=0## I found this online ##B^2 - 4AC > 0##, hyperbola ##B^2 - 4AC = 0##, parabola ##B^2 - 4AC < 0##, ellipse or circle (circle only if B = 0 and A = C) B=0, so we have to find A&C, to determine the form of the conic section. So the...
  10. C

    Find elements of a matrix such that its determinant is zero

    Thanks for all the help today! I am super sleepy and can't think anymore, so I will look at this again tomorrow.
  11. C

    Find elements of a matrix such that its determinant is zero

    No, I don't see any ##x_1x_2## terms in the determinant expansion I had previously posted. For this 4x4 matrix, we can't possibly have x_1*x_2 since we would never multiply those 2 elements in the calculation of the determinant, or more generally, any two elements from the same column in the...
  12. C

    Find elements of a matrix such that its determinant is zero

    Oh I see. I'm guessing B=0? But I am unsure why. If that's right, then we would only have linear terms.
  13. C

    Find elements of a matrix such that its determinant is zero

    Yes, quite awhile ago. I am familiar with conics as I do a numerical work with hyperbolic PDEs, but I am having a hard time tying geometrical concepts with this matrix. So the 4 terms in the last row all represent a conic section?
  14. C

    Find elements of a matrix such that its determinant is zero

    I am struggling to view this in terms of geometry. I'm not a very visual person when it comes to math. Can you explain what you mean by "your expansion is quadratic in ##x_1## and ##x_1##"?
  15. C

    Find elements of a matrix such that its determinant is zero

    I took the determinant of this matrix using matlab. Here is what I got: a1^2*b1*c2 - a1^2*b1*x2 - a1^2*b2*c1 + a1^2*b2*x1 + a1^2*c1*x2 - a1^2*c2*x1 - a1*b1^2*c2 + a1*b1^2*x2 - a1*b2^2*c2 + a1*b2^2*x2 + a1*b2*c1^2 + a1*b2*c2^2 - a1*b2*x1^2 - a1*b2*x2^2 - a1*c1^2*x2 - a1*c2^2*x2 + a1*c2*x1^2 +...
  16. C

    Find elements of a matrix such that its determinant is zero

    Is the neat way by expanding out the determinant?
  17. C

    Find elements of a matrix such that its determinant is zero

    Hello all, sorry for the late reply. I was caught up in other courses and neglected this one. I am doing this now, first using the 3 equation, 3 unknowns approach. I haven't attempted to expand, as I was intimated by a 4x4 matrix of all non-zeros.
  18. C

    Find elements of a matrix such that its determinant is zero

    Oh! I see what you are saying. We need one additional equation and that can be obtained by formulating an equation using the 1st row (i.e., 1=u+v+w) or the last column (i.e., ##x_1^2+x_2^2 = u(a_1^2+a_2^2)+v(b_1^2+b_2^2)+w(c_1^2+c_2^2)##)
  19. C

    Find elements of a matrix such that its determinant is zero

    I'm sorry I do not understand. What are you referring to as 2 equations and 3 unknowns?
  20. C

    Find elements of a matrix such that its determinant is zero

    Yes, it means that. [x1 x2] can be any scalar multiple of [a1 a2], [b1 b2], and/or [c1 c2]. Could I say that [x1 x2] is spanned by the vectors [a1 a2], [b1 b2], and[c1 c2]?
  21. C

    Find elements of a matrix such that its determinant is zero

    Homework Statement Please see the attached file if my inline insertion does not work. Homework Equations ##det(A)=det(A^T)##[/B] The Attempt at a Solution Since a matrix has a determinant of zero only when it's columns are linearly dependent, we look for a set of points [x1 x2] such...
  22. C

    QR Factorization

    Ah yes! I think I overthought this problem.
  23. C

    QR Factorization

    Homework Statement Consider an invertible n x n matrix A. Can you write A as A=LQ, where L is a lower triangular matrix and Q is orthogonal? Hint: Consider the QR factorization of #A^T#. Homework Equations For QR factorization, Q is orthogonal and R is upper triangular. The Attempt at a...
  24. C

    Find all 2x2 matrices X such that AX=XA for all 2x2 matrices

    Ah I see what you are saying. I had forgotten the essence of the problem statement. Regarding the rigorous approach, it seems there are too many unknowns for that to come out nicely?
  25. C

    Find all 2x2 matrices X such that AX=XA for all 2x2 matrices

    Hello. Thank you for the reply. The algebra for (2)-(4) were based on the results of (1). For (2), we have ax+bz=wb+xd; in (1) we decided that x=y=0, thus this expression becomes bz=wb For (3), if x=y=0, then w=z For (4), we have cx+dz=yb+zd=> xc=yb, thus x=y=0. I am not too confident with...
  26. C

    Find all 2x2 matrices X such that AX=XA for all 2x2 matrices

    Homework Statement Find all 2x2 matrices X such that AX=XA for all 2x2 matrices The Attempt at a Solution Let A = a b c d and X = w x y z Then AX = XA ==> aw+by=wa+xc .........(1) ax+bz=wb+xd .........(2) cw+dy=ya+zc .........(3) cx+dz=yb+zd .........(4) (1) ==> by = xc, which...
  27. C

    Orthogonal projection and reflection (matrices)

    Homework Statement [Imgur](http://i.imgur.com/VFT1haQ.png) Homework Equations reflection matrix = 2*projection matrix - Identity matrix The Attempt at a Solution Using the above equation, I get that B is the projection matrix and E is the reflection matrix. Can someone please verify if this...
  28. C

    Angular frequency=2*pi*frequency

    He showed us the "whole" rotation, not parts. Refer to the OP where it mentioned "axis of rotation." It's oscillating between the + and - angles.
  29. C

    Angular frequency=2*pi*frequency

    I am pretty sure that the ω=2*∏*f only applies to continuous rotations. In your setup, the angular displacement is discontinuous at the max and mins. The setups are not rotating at the same speed=>they have diff. angular frequencies.
  30. C

    Circuits with diode

    It would be VA=5V across the capacitor since no current will be flowing through this circuit at steady state. Is this the reason why the left hand side of the diode's voltage is greater than the right? Thus, creating a forward biased diode.
Top