Usng properties of determinants- prove that -beginvmatrix181 - 1-3 x1-3 y - 1 - 11 - 1-3 z - 1lend vmatrix -93 x y z-x y-y z-z x

LHS : Let Δ = 1 amp;1 amp;1+3x 1+3y amp;1 amp;1 1 amp; 1+3z amp;1 By C_1 → C_1 C_2, C_2 → C_2 C_3 ⇒Δ = 0 amp; 3x amp;1+3x ...

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Derivative of Some Standard Functions

Usng properti...

Question

Usng properties of determinants , prove that $∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 + 3 x 1 + 3 y & 1 & 1 1 & 1 + 3 z & 1 \end{matrix} ∣ ∣ ∣ ∣ = 9 (3 x y z + x y + y z + z x)$

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∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 + 3 x 1 + 3 y & 1 & 1 1 & 1 + 3 z & 1 \end{matrix} ∣ ∣ ∣ ∣ = 9 (3 x y z + x y + y z + z x)

|\begin{matrix} 1 & 1 & 1 + 3 x \\ 1 + 3 y & 1 & 1 \\ 1 & 1 + 3 z & 1 \end{matrix}| = 9 (3 x y z + x y + y z + z x) .

∣ ∣ ∣ ∣ ∣ \begin{matrix} x^{2} + 1 & x y & x z x y & y^{2} + 1 & y z x z & y z & z^{2} + 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 1 + x^{2} + y^{2} + z^{2} .

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

∣ ∣ ∣ ∣ ∣ \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)

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