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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