-a(b^2+c^2-a^2)&2b^3&2c^3 2a^3&-b(c^2+a^2-b^2)&2c^3 2a^3&2b^3&-c(a^2+b^2-c^2) =abc(a^2+b^2+c^2)

-a(b^2+c^2-a^2)&2b^3&2c^3 2a^3&-b(c^2+a^2-b^2)&2c^3 2a^3&2b^3&-c(a^2+b^2-c^2) =abc(a^2+b^2+c^2) L.H.S= -a(b^2+c^2-a^2)&2b^3&2c^3 2a^3&-b(c^2+a^2-b^2)&2c^3 2a^3&2b^3&-c(a^2+b^2-c^2) Tkae a, b and c common from C_1, C_2 and C_3 respectively. =abc -(b^2+c^2-a^2)&2b^2&2c^2 2a^2&-(c^2+a^2-b^2)&2c^2 2a^2&2b^2&-(a^2+b^2-c^2) Apply: R_1→ R_1- R_3, R_2→ R_2- R_3 =abc -(b^2+c^2-a^2)-2a^2&0&2c^2+(a^2+b^2-c^2) 0&-(c^2+a^2-b^2)-2b^2&2c^2+(a^2+b^2-c^2) 2a^2&2b^2&-(a^2+b^2-c^2) =abc -(b^2+c^2-a^2)-2a^2&0&(a^2+b^2-c^2) 0&-(c^2+a^2-b^2)-2b^2&2c^2+(a^2+b^2-c^2) 2a^2&2b^2&-(a^2+b^2-c^2) =abc(b^2+c^2+a^2)^2 -1&0&1 0&-1&1 2a^2&2b^2&-(a^2+b^2-c^2) =abc(b^2+c^2+a^2)^2 -1&0&1 0&-1&1 2a^2&2b^2&-(a^2+b^2-c^2)+2a^2 =abc(b^2+c^2+a^2)^2 -1&0&1 0&-1&1 2a^2&2b^2&-b^2+c^2+a^2 =-abc(b^2+c^2+a^2)[(-1)(-b^2+c^2+a^2)-(1)(2b^2)] =abc(a^2+b^2+c^2)^3 =R.H.S

-a(b^2+c^2-a^2)&2b^3&2c^3 2a^3&-b(c^2+a^2-b^2)&2c^3 2a^3&2b^3&-c(…

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$\begin{vmatrix}-\text{a}(\text{b}^2+\text{c}^2-\text{a}^2)&2\text{b}^3&2\text{c}^3\\2\text{a}^3&-\text{b}(\text{c}^2+\text{a}^2-\text{b}^2)&2\text{c}^3\\2\text{a}^3&2\text{b}^3&-\text{c}(\text{a}^2+\text{b}^2-\text{c}^2)\end{vmatrix}$
$=\text{abc}(\text{a}^2+\text{b}^2+\text{c}^2)$
$\text{L.H.S}=\begin{vmatrix}-\text{a}(\text{b}^2+\text{c}^2-\text{a}^2)&2\text{b}^3&2\text{c}^3\\2\text{a}^3&-\text{b}(\text{c}^2+\text{a}^2-\text{b}^2)&2\text{c}^3\\2\text{a}^3&2\text{b}^3&-\text{c}(\text{a}^2+\text{b}^2-\text{c}^2)\end{vmatrix}$
Tkae $\mathrm{a}, \mathrm{b}$ and c common from $\mathrm{C}_1, \mathrm{C}_2$ and $\mathrm{C}_3$ respectively.
$=\text{abc}\begin{vmatrix}-(\text{b}^2+\text{c}^2-\text{a}^2)&2\text{b}^2&2\text{c}^2\\2\text{a}^2&-(\text{c}^2+\text{a}^2-\text{b}^2)&2\text{c}^2\\2\text{a}^2&2\text{b}^2&-(\text{a}^2+\text{b}^2-\text{c}^2)\end{vmatrix}$
Apply: $R_1→ R_1- R_3, R_2→ R_2- R_3$
$=\text{abc}\begin{vmatrix}-(\text{b}^2+\text{c}^2-\text{a}^2)-2\text{a}^2&0&2\text{c}^2+(\text{a}^2+\text{b}^2-\text{c}^2)\\0&-(\text{c}^2+\text{a}^2-\text{b}^2)-2\text{b}^2&2\text{c}^2+(\text{a}^2+\text{b}^2-\text{c}^2)\\2\text{a}^2&2\text{b}^2&-(\text{a}^2+\text{b}^2-\text{c}^2)\end{vmatrix}$
$=\text{abc}\begin{vmatrix}-(\text{b}^2+\text{c}^2-\text{a}^2)-2\text{a}^2&0&(\text{a}^2+\text{b}^2-\text{c}^2)\\0&-(\text{c}^2+\text{a}^2-\text{b}^2)-2\text{b}^2&2\text{c}^2+(\text{a}^2+\text{b}^2-\text{c}^2)\\2\text{a}^2&2\text{b}^2&-(\text{a}^2+\text{b}^2-\text{c}^2)\end{vmatrix}$
$=\text{abc}(\text{b}^2+\text{c}^2+\text{a}^2)^2\begin{vmatrix}-1&0&1\\0&-1&1\\2\text{a}^2&2\text{b}^2&-(\text{a}^2+\text{b}^2-\text{c}^2)\end{vmatrix}$
$=\text{abc}(\text{b}^2+\text{c}^2+\text{a}^2)^2\begin{vmatrix}-1&0&1\\0&-1&1\\2\text{a}^2&2\text{b}^2&-(\text{a}^2+\text{b}^2-\text{c}^2)+2\text{a}^2\end{vmatrix}$
$=\text{abc}(\text{b}^2+\text{c}^2+\text{a}^2)^2\begin{vmatrix}-1&0&1\\0&-1&1\\2\text{a}^2&2\text{b}^2&-\text{b}^2+\text{c}^2+\text{a}^2\end{vmatrix}$
$=-\text{abc}(\text{b}^2+\text{c}^2+\text{a}^2)[(-1)(-\text{b}^2+\text{c}^2+\text{a}^2)-(1)(2\text{b}^2)]$
$=\text{abc}(\text{a}^2+\text{b}^2+\text{c}^2)^3$
$=\text{R.H.S}$

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