Matrix Differentiation

Question

$\text{If } f(x,y,z)=\begin{bmatrix} x^2&y^2&z^2\\ x&y&z\\ 1&1&1 \end{bmatrix},\quad \text{prove that} fx + fy + fz = 0$

I have started the initial differential, however am not sure of the next step.

Answer 1

Hint.

I think you wrote down the question with a bit misleading notation, as also John Habert pointed out. The only way it makes sense and your equation $fx+fy+fz=0$ is fulfilled, if you mean this

$$ f(x,y,z) = \text{det} \begin{bmatrix} x^2&y^2&z^2\\ x&y&z\\ 1&1&1 \end{bmatrix},$$

and instead of $fx$, $fy$, and $fz$ we have $f_x$, $f_y$, and $f_z$ - where these latter denote partial derivatives, $f_x=\tfrac{\partial f}{\partial x}$, $f_y=\tfrac{\partial f}{\partial y}$, and $f_z=\tfrac{\partial f}{\partial z}$.

Calculating the determinant, we get $f(x,y,z)=x^2y+y^2z +z^2x - z^2 y -y^2 x -x^2z$. Now obtain $f_x$, $f_y$ and $f_z$ by differentiating this expression wrt $x$, $y$ and $z$, respectively. This should be sufficient to get the answer.