第16章 16.4

(1)

$\begin{vmatrix} a & a^2 & b+c \\ b & b^2 & c+a \\ c & c^2 & a+b \end{vmatrix} = ab^2(a+b)+a^2c(c+a)+bc^2(b+c)-b^2c(b+c)-ac^2(c+a)-a^2b(a+b)$

$= a^2b^2 + ab^3 + a^2c^2 + a^3c + b^2c^2 + bc^3 - b^3c - b^2c^2 - ac^3 - a^2c^2 - a^3b - a^2b^2$

$= ab^3 + a^3c + bc^3 - b^3c - ac^3 - a^3b = a^3(c-b) - a(b^3-c^3) - bc(b^2-c^2)$

$= -a^3(b-c) - a(b-c)(b^2+bc+c^2) - bc(b-c)(b+c)$

$= -(b-c)\{a^3 + a(b^2+bc+c^2) - bc(b+c)\}$

$= -(b-c)(a-b)(a-c)(a+b+c)$

(2)

$\begin{vmatrix} a & b & c \\ a^2 & b^2 & c^2 \\ bc & ca & ab \end{vmatrix} = a^2b^3 + b^2c^3 + a^3c^2 - b^3c^2 - a^2c^3 - a^3b^2$

$= a^3(c^2-b^2) + a^2(b^3-c^3) + b^2c^2(c-b) = a^3(c-b)(c+b) - a^2(c-b)(c^2+cb+b^2) + b^2c^2(c-b)$

$= (c-b)\{a^3(b+c) - a^2(b^2+bc+c^2) + b^2c^2\}$

$= -(b-c)(a-b)(a-c)\{a(b+c)+bc\} = (b-c)(a-b)(a-c)(ab+bc+ca)$

(3)

$\begin{vmatrix} a+b+c & -c & -b \\ -c & a+b+c & -a \\ -b & -a & a+b+c \end{vmatrix} = (a+b+c)^2 - 2abc - (a+b+c)(a^2+b^2+c^2)$

$= (a+b+c)(a^2+b^2+c^2+2ab+2bc+2ca-a^2+b^2+c^2) - 2abc$

$= (a+b+c)(2ab+2bc+2ca) - 2abc$

$= 2a^2b + 2abc + 2a^2c + 2ab^2 + 2b^2c + 2abc + 2abc + 2bc^2 + 2ac^2 - 2abc$

$= 2\{a^2(b+c) + a(b^2+c^2+2bc) + b^2c + bc^2\} = 2\{a^2(b+c) + a(b+c)^2 + bc(b+c)\}$

$= 2(b+c)\{a^2+a(b+c)+bc\} = 2(b+c)(a+b)(a+c)$