LA2.1.14

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Prove that {\begin{vmatrix}a^{2}&bc&ac+c^{2}\\a^{2}+ab&b^{2}&ac\\ab&b^{2}+bc&c^{2}\end{vmatrix}}=4a^{2}b^{2}c^{2}\,

Taking common factors from C1,C2,C3,we have

{\begin{vmatrix}a^{2}&bc&ac+c^{2}\\a^{2}+ab&b^{2}&ac\\ab&b^{2}+bc&c^{2}\end{vmatrix}}=abc{\begin{vmatrix}a&c&a+c\\a+b&b&a\\b&b+c&c\end{vmatrix}}\,

By C_{1}+C_{2}-C_{3}\, we get

abc{\begin{vmatrix}0&c&a+c\\2b&b&a\\2b&b+c&c\end{vmatrix}}\,

Applying R_{3}-R_{2}\,

abc{\begin{vmatrix}0&c&a+c\\2b&b&a\\0&c&c-a\end{vmatrix}}=abc(-2b)[c(c-a)-c(a+c)]=abc(-2b)(-2ca)=4a^{2}b^{2}c^{2}\,=RHS


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