A determinant and a surface - 3

Algebra Level pending

Classify the surface $\mathbf{r} = (x, y, z)$ , where $x$ , $y$ and $z$ satisfy the determinant equation:

$\begin{vmatrix} x && 1 && -y \\ -2 && x && z \\ z && y && 3 \end{vmatrix} = 0$

Paraboloid Hyperboloid of one sheet Hyperboloid of two sheets Ellipsoid The Empty Set A Single Point

1 solution

David Vreken
Nov 19, 2020

The determinant calculates to $3x^2 + z^2 + 2y^2 + 6 = 0$ , or after rearranging, $3x^2 + 2y^2 + z^2 = -6$ . Since $3x^2$ , $2y^2$ , and $z^2$ are all non-negative numbers, and the sum of three non-negative numbers can't be negative, there are no all real solutions to the equation and the surface is the empty set .

A determinant and a surface - 3

1 solution

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