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

X+yy+zz+xzxy 3 3 3=x+y+zx+y+zz+x+yzxy 3 3 3 #160; #160; #160; #160; #160; #160; #160;Applying #160;R1 #8594;R1+R2=x+y+z111zxy 3 3 3 #160; ...

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(x±a)(x±b) = x²±(a+b)x+ab.

Write the val...

Question

Write the value of the determinant $|\begin{matrix} x + y & y + z & z + x \\ z & x & y \\ - 3 & - 3 & - 3 \end{matrix}|$ .

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△ = ∣ ∣ ∣ ∣ \begin{matrix} x + y & y + z & z + x z & x & y - 3 & - 3 & - 3 \end{matrix} ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣ \begin{matrix} x & x^{2} & y z y & y^{2} & z x z & z^{2} & x y \end{matrix} ∣ ∣ ∣ ∣ ∣ = (x - y) (y - z) (z - x) (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})

- k

ω ∣ ∣ ∣ ∣ ∣ \begin{matrix} x & y & z x^{2} & y^{2} & z^{2} y + z & z + x & x + y \end{matrix} ∣ ∣ ∣ ∣ ∣ = (x - y) (y - z) (z - x) (x + y + z)

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

= (1 + p x y z) (x - y) (y - z) (z - x)

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