I'd like to find the determinant
of following matrix
$$ \begin{pmatrix} {x_1}^2 & x_1y_1 & {y_1}^2 & x_1 & y_1 \\ {x_2}^2 & x_2y_2 & {y_2}^2 & x_2 & y_2 \\ {x_3}^2 & x_3y_3 & {y_3}^2 & x_3 & y_3 \\ {x_4}^2 & x_4y_4 & {y_4}^2 & x_4 & y_4 \\ {x_5}^2 & x_5y_5 & {y_5}^2 & x_5 & y_5 \\ \end{pmatrix} $$
where $ x_i \not=x_j $,
$ y_i \not= y_j $ for $ i \not= j $
($ i,j=1,2,3,4,5$)
And I'd like to verify the determinant is not zero.
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I was wondering this while studying
Vandermonde matrix, conic section - five points ... and so on.
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By the way,
could recommend any program or website which calculate determinant of matrix?
Actually I touched "wolframalpha.com", but wolfram can't recognize my input. (maybe I did mistake...)
$$ $$ Thank you for your attention to this matter.
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EDIT (ADD)
The conic section equation will be of the form
$$ Ax^2 + Bxy+Cy^2 +D x+Ey +F= 0 --- (*) $$with A, B, C not all zero
If the points $(x_1, y_1), (x_2, y_2), \cdots, (x_5, y_5)$ satisfy (*) , then
$$ \begin{pmatrix} {x_1}^2 & x_1y_1 & {y_1}^2 & x_1 & y_1 \\ {x_2}^2 & x_2y_2 & {y_2}^2 & x_2 & y_2 \\ {x_3}^2 & x_3y_3 & {y_3}^2 & x_3 & y_3 \\ {x_4}^2 & x_4y_4 & {y_4}^2 & x_4 & y_4 \\ {x_5}^2 & x_5y_5 & {y_5}^2 & x_5 & y_5 \\ \end{pmatrix}\begin{pmatrix} A \\ B \\ C \\ D \\ E \\ \end{pmatrix} = \begin{pmatrix} -F \\ -F \\ -F \\ -F \\ -F \\ \end{pmatrix} $$
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BUT in this time
I know the fact(statement) "five points decide conic section UNIQUELY"
So, I thought the determinant of matrix will be nonzero
(and I asked you guys that 'convinced me the determinant is nonzero'...)
BUT the determinant is zero...
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Where I did mistake?
PLEASE help me. Thanks a lot.