Word Count : 940

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We are given $$ \begin{vmatrix} b+c & c+a & a+b\\[0.5em] c+a & a+b & b+c\\[0.5em] a+b & b+c & c+a \end{vmatrix} = 2 \begin{vmatrix} a & b & c\\[0.5em] b & c & a\\[0.5em] c & a & b \end{vmatrix} $$ --- First compute the determinant on the right: $$ \Delta_R= \begin{vmatrix} a & b & c\\ b & c & a\\ c & a & b \end{vmatrix} $$ Use the cofactor expansion along the first row: $$ \Delta_R = a\begin{vmatrix} c & a \\ a & b \end{vmatrix} - b\begin{vmatrix} b & a \\ c & b \end{vmatrix} + c\begin{vmatrix} b & c \\ c & a \end{vmatrix} $$ Compute each: $$ a(c b - a a) - b(b b - a c) + c(b a - c c) $$ $$ = a(cb - a^2) - b(b^2 - ac) + c(ab - c^2) $$ $$ = a c b - a^3 - b^3 + a b c + a b c - c^3 $$ Collect: $$ \Delta_R = a b c + a b c + a b c - (a^3 + b^3 + c^3) $$ $$ \Delta_R = 3abc - (a^3 + b^3 + c^3) $$ --- Now compute the determinant on the left: $$ \Delta_L = \begin{vmatrix} b+c & c+a & a+b\\ c+a & a+b & b+c\\ a+b & b+c & c+a \end{vmatrix} $$ Write each entry in terms of $a,b,c$ sums. __________ ____ ________ ______ ________ _________ __________ __________ ________.
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