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Consistency of Linear System of Equations

If f=x+y+z; g...

Question

$I f f = x + y + z; g = x^{2} + y^{2} + z^{2}; h = x^{3} + y^{3} + z^{3}$ ,

then $∣ ∣ ∣ ∣ ∣ \begin{matrix} f_{x} & f_{y} & f_{z} g_{x} & g_{y} & g_{z} h_{x} & h_{y} & h_{z} \end{matrix} ∣ ∣ ∣ ∣ ∣ = ?$

A

- 1

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B

0

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C

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D

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

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Solution

The correct option is B $6 (x - y) (y - z) (z - x)$
$∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 2 x & 2 y & 2 z 3 x^{2} & 3 y^{2} & 3 z^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 6 ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 x & y & z x^{2} & y^{2} & z^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 6 ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 0 x & y - x & z - x x^{2} & y^{2} - x^{2} & z^{2} - x^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= 6 (z - x) (y - x) ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 0 x & 1 & 1 x^{2} & (y - x) & (z + x) \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= 6 (z - x) (y - x) (z - y)$

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Consistency of Linear System of Equations

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