If x-y-z-0-find - x y z x 2 y 2 z 2 y-z z-x x-y - -

Consider the following equation .given that,(x+y+z)=0 #160; | x amp; y amp; z x ^ 2 amp; y^ 2 amp; z ^ 2 y+z amp; z+x amp; ...

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Properties of Determinants

If x+y+z=0,...

Question

If $x + y + z = 0$ ,find
$∣ ∣ ∣ ∣ ∣ \begin{matrix} x & y & z x^{2} & y^{2} & z^{2} y + z & z + x & x + y \end{matrix} ∣ ∣ ∣ ∣ ∣ =$

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Solution

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Similar questions

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)

\frac{x}{x + 2 y + z} = \frac{y}{y + 2 z + x} = \frac{z}{z + 2 x + y}

\neq

\frac{1}{4}

(y - z) (z + x) = 22

(z + x) (x - y) = 33

(x - y) (y - z) = 6

Prove that:
$∣ ∣ ∣ ∣ ∣ \begin{matrix} y^{2} z^{2} & y z & y + z z^{2} x^{2} & z x & z + x x^{2} y^{2} & x y & x + y \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$

a + b + c = 0

x + y + z = 0

4 {(a x + b y + c z)}^{3} - 3 (a x + b y + c z) (a^{2} + b^{2} + c^{2}) (x^{2} + y^{2} + z^{2}) - 2 (b - c) (c - a) (a - b)

(y - z) (z - x) (x - y) = 54 a b c x y z

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