How to determine this 4x4 matrix?

[tomnomath]

Definition of S

S(a,b,c)≡(a+b+c)2−2(a2+b2+c2)−4abcS(a,b,c)≡(a+b+c)2−2(a2+b2+c2)−4abc​

So if I expand this I get

S(a,b,c)=2ab+2ac+2bc−a2−b2−c2−4abcS(a,b,c)=2ab+2ac+2bc−a2−b2−c2−4abc
=4(1−a)(1−b)(1−c)−(a+b+c−2)2=4(1−a)(1−b)(1−c)−(a+b+c−2)2
=−det\begin{matrix}0&a&b&1\\a&0&c&1\\b&c&0&1\\1&1&1&2\end{matrix}​

Using row echelon form I checked that the negative of the determinant of the matrix is equal to the first equation that resulted from my expanding the original SS. I have yet to make a determination on the second equation. But I realized that I do not know where this matrix came from. By this I mean, I do not know how I would set up the matrix. For instance I can setup the matrix for say three points, $a_1,a_2,a_3$ where $a_1\equiv [x_1,y_1,z_1] etc such that each row of the 3x3 matrix is comprised of the components of $a_1,a_2,a_3$ such that the 3x3 matrix looks like:

\begin{matrix}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{matrix}
But I do not have the same clarity as to how to setup the 4x4 matrix. Any suggestions or links that could give me guidance as to how to setup this 4x4 matrix given that all I would have is the equations I listed above.​

 

[stapel]

I don't know what you're seeing on your screen, but I suspect that nobody has attempted a response because of how poorly the text is formatted on our end. The following is what I think you meant to say:

S is defined by:

. . . . .\(\displaystyle S(a,\, b,\, c)\, \equiv\, (a\, +\, b\, +\, c)^2\, -\, 2\, (a^2\, +\, b^2\, +\, c^2)\, -\, 4abc\)

I'm guessing the above is the expression for part of an exercise. No instructions were provided. You then said:

If I expand the right-hand side of the definition, I get the following:

. . . . .\(\displaystyle \begin{align}S(a,\, b,\, c)\, &=\, 2ab\, +\, 2ac\, +\, 2ac\, +\, 2bc\, -\, a^2\, -\, b^2\, -\, c^2\, -\, 4abc

\\ \\ &=\, 4\, (1\, -\, a)\, (1\, -\, b)\, (1\, -\, c)\, -\, (a\, +\, b\, +\, c\, -\, 2)^2

\\ \\ &=\, -\left|\begin{array}{cccc}0&a&b&1\\a&0&c&1\\b&c&0&1\\1&1&1&2\end{array}\right|\end{align}\)

Using row echelon form I checked that the negative of the determinant of the matrix is equal to the first equation that resulted from my expanding the original SS.

What do you mean by "the original SS"? How does "SS" relate to the defined "S"?

I have yet to make a determination on the second equation.

What is "the second equation"? What is the "determination" that you're supposed to be making? Why?

But I realized that I do not know where this matrix came from. By this I mean, I do not know how I would set up the matrix. For instance I can setup the matrix for say three points,

. . . . .\(\displaystyle a_1,\,a_2,\,a_3\)

where

. . . . .\(\displaystyle a_1\, \equiv [x_1,\,y_1,\,z_1]\)

etc...

From whence are these new variables coming? How do they relate to anything that has come before?

...such that each row of the 3x3 matrix is comprised of the components of

. . . . .\(\displaystyle a_1,\,a_2,\,a_3\)

such that the 3x3 matrix looks like:

. . . . .\(\displaystyle \left[\begin{array}{ccc}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{array}\right]\)

But I do not have the same clarity as to how to setup the 4x4 matrix. Any suggestions or links that could give me guidance as to how to setup this 4x4 matrix given that all I would have is the equations I listed above.

Since we have no idea what you're supposed to be doing, what the various equations are, what might be the source of the new variables or the need for a new matrix, I'm afraid there is little we can do.

Kindly please reply with the rest of the information for this exercise. Thank you!