Prove that: -bc&b^2+bc&c^2+bc ^2+ac&-ac&c^2+ac ^2+ab&b^2+ab&-ab =(ab+bc+ca)^3

L.H.S= -bc&b^2+bc&c^2+bc ^2+ac&-ac&c^2+ac ^2+ab&b^2+ab&-ab Multiply R1, R2 and R3 by a, b and c respectively. =1 abc -abc&ab^2+abc&ac^2+abc ^2b+abc&-abc&bc^2+abc ^2c+abc&b^2c+abc&-abc Take a, b and c common from C1, C2 and C3 respectively. = abc abc -bc&ab+ac&ac+ab +bc&-ac&bc+ab +bc&bc+ac&-ab Apply: R1 → R1 + R2 + R3 = ab+bc+ca&ab+bc+ca&ab+bc+ca +bc&-ac&bc+ab +bc&bc+ac&-ab =(ab+bc+ca) 1&1&1 +bc&-ac&bc+ab +bc&bc+ac&-ab =(ab+bc+ca) 0&1&0 +bc+ac&-ac&bc+ab+ac 0&bc+ac&-ab-bc-ac =(ab+bc+ca)^3 0&1&0 0&-ac&1 0&bc+ac&1 =(ab+bc+ca)^3 =R.H.S

Prove that: -bc&b^2+bc&c^2+bc ^2+ac&-ac&c^2+ac ^2+ab&b^2+ab&-ab =…

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Question

Prove that:
$\begin{vmatrix}-\text{bc}&\text{b}^2+\text{bc}&\text{c}^2+\text{bc}\\\text{a}^2+\text{ac}&-\text{ac}&\text{c}^2+\text{ac}\\\text{a}^2+\text{ab}&\text{b}^2+\text{ab}&-\text{ab}\end{vmatrix}$
$=(\text{ab}+\text{bc}+\text{ca})^3$

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Answer

$\text{L.H.S}=\begin{vmatrix}-\text{bc}&\text{b}^2+\text{bc}&\text{c}^2+\text{bc}\\\text{a}^2+\text{ac}&-\text{ac}&\text{c}^2+\text{ac}\\\text{a}^2+\text{ab}&\text{b}^2+\text{ab}&-\text{ab}\end{vmatrix}$
Multiply R₁, R₂ and R₃ by a, b and c respectively.
$=\frac{1}{\text{abc}}\begin{vmatrix}-\text{abc}&\text{ab}^2+\text{abc}&\text{ac}^2+\text{abc}\\\text{a}^2\text{b}+\text{abc}&-\text{abc}&\text{bc}^2+\text{abc}\\\text{a}^2\text{c}+\text{abc}&\text{b}^2\text{c}+\text{abc}&-\text{abc}\end{vmatrix}$
Take a, b and c common from C₁, C₂ and C₃ respectively.
$=\frac{\text{abc}}{\text{abc}}\begin{vmatrix}-\text{bc}&\text{ab}+\text{ac}&\text{ac}+\text{ab}\\\text{a}\text{b}+\text{bc}&-\text{ac}&\text{bc}+\text{ab}\\\text{a}\text{c}+\text{bc}&\text{b}\text{c}+\text{ac}&-\text{ab}\end{vmatrix}$
Apply: R₁ → R₁ + R₂ + R₃
$=\begin{vmatrix}\text{ab}+\text{bc}+\text{ca}&\text{ab}+\text{bc}+\text{ca}&\text{ab}+\text{bc}+\text{ca}\\\text{a}\text{b}+\text{bc}&-\text{ac}&\text{bc}+\text{ab}\\\text{a}\text{c}+\text{bc}&\text{b}\text{c}+\text{ac}&-\text{ab}\end{vmatrix}$
$=(\text{ab}+\text{bc}+\text{ca})\begin{vmatrix}1&1&1\\\text{a}\text{b}+\text{bc}&-\text{ac}&\text{bc}+\text{ab}\\\text{a}\text{c}+\text{bc}&\text{b}\text{c}+\text{ac}&-\text{ab}\end{vmatrix}$
$=(\text{ab}+\text{bc}+\text{ca})\begin{vmatrix}0&1&0\\\text{a}\text{b}+\text{bc}+\text{ac}&-\text{ac}&\text{bc}+\text{ab}+\text{ac}\\0&\text{b}\text{c}+\text{ac}&-\text{ab}-\text{bc}-\text{ac}\end{vmatrix}$
$=(\text{ab}+\text{bc}+\text{ca})^3\begin{vmatrix}0&1&0\\0&-\text{ac}&1\\0&\text{b}\text{c}+\text{ac}&1\end{vmatrix}$
$=(\text{ab}+\text{bc}+\text{ca})^3$
$=\text{R.H.S}$