Recent content by Clandry

1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 +...