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