Beginvmatrixx-4 - 2 x - 2 x2 x - x-4 - 2 x2 x - 2 x - x-4Lend vmatrix -5 x-44-x - 2 beginvmatrix y-k - y - x yy - y-k - y y - y - y-k

LHS= x+4 amp;2x amp;2x 2x amp; x+4 amp;2x 2x amp;2x amp;x+4 = 5x+4 amp; 2x amp; 2x 5x+4 amp;x+4 amp;2x 5x+4 amp; ...

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Consistency of Linear System of Equations

beginvmatrixx...

Question

$∣ ∣ ∣ ∣ \begin{matrix} x + 4 & 2 x & 2 x 2 x & x + 4 & 2 x 2 x & 2 x & x + 4 \end{matrix} ∣ ∣ ∣ ∣ = (5 x + 4) (4 - x)^{2}$ .

$∣ ∣ ∣ ∣ \begin{matrix} y + k & y & y y & y + k & y y & y & y + k \end{matrix} ∣ ∣ ∣ ∣ = k^{2} (3 y + k)$

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∣ ∣ ∣ ∣ \begin{matrix} x + 4 & 2 x & 2 x 2 x & x + 4 & 2 x 2 x & 2 x & x + 4 \end{matrix} ∣ ∣ ∣ ∣ = (5 x + 4) (4 - {x)}^{2}

∣ ∣ ∣ ∣ \begin{matrix} y + k & y & y y & y + k & y y & y & y + k \end{matrix} ∣ ∣ ∣ ∣ = k^{2} (3 y + k)

$∣ ∣ ∣ ∣ \begin{matrix} x + y + 2 z & x & y z & y + z + 2 x & y z & x & z + x + 2 y \end{matrix} ∣ ∣ ∣ ∣ = 2 (x + y + z)^{3}$

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

k

Find the real part of $(\frac{(x + i y) + i}{(x + i y) + 2})$
Take $(| (x + 2) + i y |)$ = $\sqrt{\frac{1}{k}}$

∣ ∣ ∣ ∣ ∣ \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})

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