LA2.1.9

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Show that {\begin{vmatrix}a+b&b+c&c+a\\b+c&c+a&a+b\\c+a&a+b&b+c\end{vmatrix}}=2{\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}}\,

LHS={\begin{vmatrix}a+b&b+c&c+a\\b+c&c+a&a+b\\c+a&a+b&b+c\end{vmatrix}}\,

Applying C_{1}\rightarrow C_{1}+C_{2}+C_{3}\,

LHS={\begin{vmatrix}2(a+b+c)&b+c&c+a\\2(a+b+c)&c+a&a+b\\2(c+a+b)&a+b&b+c\end{vmatrix}}=2{\begin{vmatrix}a+b+c&a+b+c&c+a\\b+c&c+a&a+b\\a+b+c&a+b&b+c\end{vmatrix}}\,

Applying C_{1}\rightarrow C_{1}-C_{2}\,

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

Applying C_{3}\rightarrow C_{3}-C_{1}\,

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

Applying C_{2}\rightarrow C_{2}-C_{3}\,

2{\begin{vmatrix}a&b&c\\b&c&a\\c&a&b\end{vmatrix}}\,=RHS


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