Word Count : 548

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We are asked to prove numerically (manually, not using Python) that: $$ \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = (a + b + c)(ab + bc + ca - a^2 - b^2 - c^2) $$ --- ### Step 1: Let’s evaluate the determinant directly We compute the determinant using cofactor expansion (expansion along the first row): $$ \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = a \cdot \begin{vmatrix} c & a \\ a & b \end{vmatrix} - b \cdot \begin{vmatrix} b & a \\ _______ _____ ______ _____ ____ ____ _________ _____ __________ ________ _____ ________.
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