Using properties of determinants prove that: [(a+b+c, -c,-b),(-c,a+b+c,-a),(-b,-a, a+b+c)] = 2(a+b)(b+c)(c+a)

Question

Using properties of determinants prove that:

\(\begin{bmatrix} a+b+c & -c & -b \\[0.3em] -c& a+b+c & -a\\[0.3em] -b & -a& a+b+c \end{bmatrix}\)= 2 (a+b) (b+c) (c+a)

Answer 1

\(\begin{bmatrix} a+b+c & -c & -b \\[0.3em] -c& a+b+c & -a\\[0.3em] -b & -a& a+b+c \end{bmatrix}\)

= (a + b)(b + c){0 + 2( - b + a + b + c) + 0}[expansion by first row] 

= 2(a + b)(b + c)(c + a)