Write the value of - x-y y-z z-x z x y -3 -3 -3-

On expansion, we get, #160; x+y amp; y+z amp; z+x z amp; x amp; y 3 amp; 3 amp; 3 =(x+y)( 3x+3y) (y+z)( 3z+3y)+(z+x)( 3z+ ...

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Determinant

Write the val...

Question

Write the value of $△ = ∣ ∣ ∣ ∣ \begin{matrix} x + y & y + z & z + x z & x & y - 3 & - 3 & - 3 \end{matrix} ∣ ∣ ∣ ∣$ .

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|\begin{matrix} x + y & y + z & z + x \\ z & x & y \\ - 3 & - 3 & - 3 \end{matrix}|

∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{k} & x^{k + 2} & x^{k + 3} y^{k} & y^{k + 2} & y^{k + 3} z^{k} & z^{k + 2} & z^{k + 3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = (x - y) (y - z) (z - x) (\frac{1}{x} + \frac{1}{y} + \frac{1}{z})

- k

∣ ∣ ∣ ∣ \begin{matrix} ^{x} C_{1} & ^{x} C_{2} & ^{x} C_{3}^{y} C_{1} & ^{y} C_{2} & ^{y} C_{3}^{z} C_{1} & ^{z} C_{2} & ^{z} C_{3} \end{matrix} ∣ ∣ ∣ ∣

= \frac{x y z}{C} (x - y) (y - z) (z - x)

∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{k} & x^{k + 2} & x^{k + 3} y^{k} & y^{k + 2} & y^{k + 3} z^{k} & z^{k + 2} & z^{k + 3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = (x - y) (y - z) (z - x) (\frac{1}{x} + \frac{1}{y} + \frac{1}{z})

\frac{50 x y z (x + y) (y + z) (z + x)}{100 x y (x + y) (y + z)}

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