Question

\( \left|\begin{array}{ccc}x & y & z \\ x^{2} & y^{2} & z^{2} \\ x^{3} & y^{3} & z^{3}\end{array}\right|=x y z(x-y)(y-z)(z-x) \)

Answer 1

 \(\begin{vmatrix}x&y&z\\x^2&y^2&z^2\\x^3&y^3&z^3\end{vmatrix}\)

Applying C₁ → \(\frac{C_1}x\), C₂ → \(\frac{C_2}y\) & C₃ → \(\frac{C_3}z\)

\(=xyz\begin{vmatrix}1&1&1\\x&y&z\\x^2&y^2&z^2\end{vmatrix}\)

Applying C₁ → C₁ - C₂  and C₂ → C₂ - C₃

\(=xyz\begin{vmatrix}0&0&1\\x-y&y-z&z\\x^2-y^2&y^2-z^2&z^2\end{vmatrix}\)

\(=xyz\begin{vmatrix}0&0&1\\x-y&y-z&z\\(x-y)(x+y)&(y+z)(y-z)&z^2\end{vmatrix}\)

Applying C₁ → \(\frac{C_1}{x-y}\) and C₂ → \(\frac{C_2}{y-z}\)

= xyz(x - y)(y - z)\(\begin{vmatrix}0&0&1\\1&1&z\\x+y&y+z&z^2\end{vmatrix}\)

= xyz(x - y)(y - z)\(\begin{vmatrix}1&1\\x+y&y+z\end{vmatrix}\) (By Expanding determinant along R₁)

= xyz(x - y) (y - z) (y + z - (x + y))

= xyz(x - y) (y - z) (z - x)