Prove that: (b+c)^2&a^2&bc c+a)^2&b^2&ca a+b)^2&c^2&ab =(a-b)(b-c)(c-b)(a+b+c)(a^2+b^2+c^2)

L.H.S= (b+c)^2&a^2&bc c+a)^2&b^2&ca a+b)^2&c^2&ab = (b+c)^2-(c+a)^2&a^2-b^2&bc-ca c+a)^2-(a+b)^2&b^2-c^2&ca-ab a+b)^2&c^2&ab [Applying R1 → R1 - R2 and R2 → R2 - R1] = (b+c)(b+2c+a)&(a+b)(a-b)&c(b-a) c-a)(b+2a+c)&(b-c)(b+c)&a(c-b) a+b)^2&c^2&ab =(a-b)(b-c) -(b+2c+a)&a+b&-c -(b+2a+c)&b+c&-a a+b)^2&c^2&ab [Applying x2 - y2 = (x + y)(x - y) and taking out (a - b) common from R1 and (b - c) from R2] =(a-b)(b-c) -2(b+c+a)&a+b&-c -2(b+a+c)&b+c&-a a+b)^2-c^2&c^2&ab [Applying C1 → C1 - C2] =(a-b)(b-c) -2(b+c+a)&a+b&-c -2(b+a+c)&b+c&-a a+b+c)(a+b-c)&c^2&ab [Applying x2 - y2 = (x + y)(x - y) in C1] =(a-b)(b-c)(a+b+c) -2&a+b&-c -2&b+c&-a a+b-c)&c^2&ab [Taking out (a + b + c) common from C1] =(a-b)(b-c)(a+b+c) -2&a+b&-c 0&c-a&c-a a+b-c)&c^2&ab [Applying R2 → R2 - R1] =(a-b)(b-c)(a+b+c)(c-a) -2&a+b&-c 0&1&1 a+b-c)&c^2&ab [Taking out (c - a) common from R2] =(a-b)(b-c)(a+b+c)(c-a) -2&a+b+c&-c 0&0&1 a+b-c)&c^2-ab&ab [Applying C2 → C2 - C3] =(a-b)(b-c)(a+b+c)(c-a) (-1) -2&a+b+c& a+b-c)&c^2-ab [Expanding along R2] =-(a-b)(b-c)(a+b+c)(c-a) -2c^2+2ab-a^2-b^2-2ab+c^2 =-(a-b)(b-c)(a+b+c)(c-a)(-a^2-b^2-c^2) =(a-b)(b-c)(c-b)(a+b+c)(a^2+b^2+c^2) =R.H.S

Prove that: (b+c)^2&a^2&bc c+a)^2&b^2&ca a+b)^2&c^2&ab =(a-b)(b-c…

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Question

Prove that:
$\begin{vmatrix}(\text{b}+\text{c})^2&\text{a}^2&\text{bc}\$\text{c}+\text{a})^2&\text{b}^2&\text{ca}\$\text{a}+\text{b})^2&\text{c}^2&\text{ab}\end{vmatrix}$
$=(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{b})(\text{a}+\text{b}+\text{c})(\text{a}^2+\text{b}^2+\text{c}^2)$

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Answer

$\text{L.H.S}=\begin{vmatrix}(\text{b}+\text{c})^2&\text{a}^2&\text{bc}\$\text{c}+\text{a})^2&\text{b}^2&\text{ca}\$\text{a}+\text{b})^2&\text{c}^2&\text{ab}\end{vmatrix}$
$=\begin{vmatrix}(\text{b}+\text{c})^2-(\text{c}+\text{a})^2&\text{a}^2-\text{b}^2&\text{bc}-\text{ca}\$\text{c}+\text{a})^2-(\text{a}+\text{b})^2&\text{b}^2-\text{c}^2&\text{ca}-\text{ab}\$\text{a}+\text{b})^2&\text{c}^2&\text{ab}\end{vmatrix}$ [Applying R₁ → R₁ - R₂ and R₂ → R₂ - R₁]
$=\begin{vmatrix}(\text{b}+\text{c})(\text{b}+2\text{c}+\text{a})&(\text{a}+\text{b})(\text{a}-\text{b})&\text{c}(\text{b}-\text{a})\$\text{c}-\text{a})(\text{b}+2\text{a}+\text{c})&(\text{b}-\text{c})(\text{b}+\text{c})&\text{a}(\text{c}-\text{b})\$\text{a}+\text{b})^2&\text{c}^2&\text{ab}\end{vmatrix}$
$=(\text{a}-\text{b})(\text{b}-\text{c})\begin{vmatrix}-(\text{b}+2\text{c}+\text{a})&\text{a}+\text{b}&-\text{c}\\-(\text{b}+2\text{a}+\text{c})&\text{b}+\text{c}&-\text{a}\$\text{a}+\text{b})^2&\text{c}^2&\text{ab}\end{vmatrix}$
[Applying x₂ - y₂ = (x + y)(x - y) and taking out (a - b) common from R₁ and (b - c) from R₂]
$=(\text{a}-\text{b})(\text{b}-\text{c})\begin{vmatrix}-2(\text{b}+\text{c}+\text{a})&\text{a}+\text{b}&-\text{c}\\-2(\text{b}+\text{a}+\text{c})&\text{b}+\text{c}&-\text{a}\$\text{a}+\text{b})^2-\text{c}^2&\text{c}^2&\text{ab}\end{vmatrix}$ [Applying C₁ → C₁ - C₂]
$=(\text{a}-\text{b})(\text{b}-\text{c})\begin{vmatrix}-2(\text{b}+\text{c}+\text{a})&\text{a}+\text{b}&-\text{c}\\-2(\text{b}+\text{a}+\text{c})&\text{b}+\text{c}&-\text{a}\$\text{a}+\text{b}+\text{c})(\text{a}+\text{b}-\text{c})&\text{c}^2&\text{ab}\end{vmatrix}$
[Applying x² - y² = (x + y)(x - y) in C₁]
$=(\text{a}-\text{b})(\text{b}-\text{c})(\text{a}+\text{b}+\text{c})\begin{vmatrix}-2&\text{a}+\text{b}&-\text{c}\\-2&\text{b}+\text{c}&-\text{a}\$\text{a}+\text{b}-\text{c})&\text{c}^2&\text{ab}\end{vmatrix}$
[Taking out (a + b + c) common from C₁]
$=(\text{a}-\text{b})(\text{b}-\text{c})(\text{a}+\text{b}+\text{c})\begin{vmatrix}-2&\text{a}+\text{b}&-\text{c}\\0&\text{c}-\text{a}&\text{c}-\text{a}\$\text{a}+\text{b}-\text{c})&\text{c}^2&\text{ab}\end{vmatrix}$
[Applying R₂ → R₂ - R₁]
$=(\text{a}-\text{b})(\text{b}-\text{c})(\text{a}+\text{b}+\text{c})(\text{c}-\text{a})\begin{vmatrix}-2&\text{a}+\text{b}&-\text{c}\\0&1&1\$\text{a}+\text{b}-\text{c})&\text{c}^2&\text{ab}\end{vmatrix}$
[Taking out (c - a) common from R₂]
$=(\text{a}-\text{b})(\text{b}-\text{c})(\text{a}+\text{b}+\text{c})(\text{c}-\text{a})\begin{vmatrix}-2&\text{a}+\text{b}+\text{c}&-\text{c}\\0&0&1\$\text{a}+\text{b}-\text{c})&\text{c}^2-\text{ab}&\text{ab}\end{vmatrix}$
[Applying C₂ → C₂ - C₃]
$=(\text{a}-\text{b})(\text{b}-\text{c})(\text{a}+\text{b}+\text{c})(\text{c}-\text{a}) \left\{(-1)\begin{vmatrix}-2&\text{a}+\text{b}+\text{c}&\$\text{a}+\text{b}-\text{c})&\text{c}^2-\text{ab}\end{vmatrix}\right\}$ [Expanding along R₂]
$=-(\text{a}-\text{b})(\text{b}-\text{c})(\text{a}+\text{b}+\text{c})(\text{c}-\text{a})\{-2\text{c}^2+2\text{ab}-\text{a}^2-\text{b}^2-2\text{ab}+\text{c}^2\}$
$=-(\text{a}-\text{b})(\text{b}-\text{c})(\text{a}+\text{b}+\text{c})(\text{c}-\text{a})(-\text{a}^2-\text{b}^2-\text{c}^2)$
$=(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{b})(\text{a}+\text{b}+\text{c})(\text{a}^2+\text{b}^2+\text{c}^2)$
$=\text{R.H.S}$