[Math] Proving two determinants are equal

Question

Using determinant properties (without expanding), prove that

$$ \begin{vmatrix}yz & z^2 & y^2 \\ z^2 & xz & x^2 \\ y^2 & x^2 & xy \end{vmatrix} =
xyz\begin{vmatrix}x & z & y \\ z & y & x \\ y & x &z\end{vmatrix}$$

I'm completely lost and I don't know what to try. Any hints?

Thanks in advance.

Answer 1

Start from

$$\begin{vmatrix}x & z & y \\ z & y & x \\ y & x &z\end{vmatrix}.$$

Multiply the colums by $y, x, \frac{xy}z$; multiply the rows by $\frac{z}{x},\frac z y, 1$. You have now multiplied the determinant with $xyz$.

Edit: More symmetric: multiply the colums $yz, zx, xy$; multiply the rows by $\frac1x,\frac1y, \frac1z$.