If $\left| \begin{matrix} x^{n} & x^{n + 2} & x^{n + 3} \ y^{n} & y^{n + 2} & y^{n + 3} \ z^{n} & z^{n + 2} & z^{n + 3} \end{matrix} ight| = (y - z)(z - x)(x - y)$ $\left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} ight),\$then $n$ is equal to

Question

If $\left| \begin{matrix} x^{n} & x^{n + 2} & x^{n + 3} \ y^{n} & y^{n + 2} & y^{n + 3} \ z^{n} & z^{n + 2} & z^{n + 3} \end{matrix} ight| = (y - z)(z - x)(x - y)$ $\left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} ight),\$then $n$ is equal to

Dr. Abubacker · Accepted Answer

(c)
Given, $\left| \begin{matrix}
x^{n} & x^{n + 2} & x^{n + 3} \
y^{n} & y^{n + 2} & y^{n + 3} \
z^{n} & z^{n + 2} & z^{n + 3}
\end{matrix} ight|$
$= (y - z)(z - x)(x - y)\left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} ight)$
The degree of determinant
$= n + (n + 2) + (n + 3) = 3n + 5$
and the degree of RHS=2
$	herefore 3n + 5 = 2 \Rightarrow n = - 1$

Q. If $\left| \begin{matrix} x^{n} & x^{n + 2} & x^{n + 3} \\ y^{n} & y^{n + 2} & y^{n + 3} \\ z^{n} & z^{n + 2} & z^{n + 3} \end{matrix} \right| = (y - z)(z - x)(x - y)$ $\left( \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \right),\$then $n$ is equal to

Explanation

Reviewed by Dr. Abubacker

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