Prove that - - 1 1 1- a b c- a3 b3 c3 - - 1 1 1- a2 b2 c2- bc ca ab -

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Properties of Determinants

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Prove that $∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣$ =$[\left| {\begin{array}{*{20}{c}}1&1&1\\{{a^2}}&{{b^2}}&{{c^2}}\\{bc}&{ca}&{ab}\end{array}} \right$

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∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 a & b & c a^{3} & b^{3} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a) (a + b + c)

Prove that $∣ ∣ ∣ ∣ \begin{matrix} a & b & c a^{2} & b^{2} & c^{2} b c & c a & a b \end{matrix} ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a) (a b + b c + c a)$

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∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & 1 α + β & α^{2} + β^{2} & 2 α β α + β & 2 α β & α^{2} + β^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣

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∣ ∣ ∣ ∣ \begin{matrix} 1 & 0 & 0 0 & α & β 0 & β & α \end{matrix} ∣ ∣ ∣ ∣

D_{2}

D_{2} = \frac{D}{D_{1}}

∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣

Δ = ∣ ∣ ∣ ∣ \begin{matrix} a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} a_{3} & b_{3} & c_{3} \end{matrix} ∣ ∣ ∣ ∣

A_{2}

B_{2}

C_{2}

a_{2}, b_{2}, c_{2}

a_{1} A_{2} + b_{1} B_{2} + c_{1} C_{2}

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