第16章 16.2

(1)

$\begin{vmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{vmatrix} = (-a)\cdot(-c)\cdot b + a\cdot c\cdot(-b) = abc - abc = 0$

(2)

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

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

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

$= 4abc$

(3)

$\begin{vmatrix} 1 & a & b+c \\ 1 & b & c+a \\ 1 & c & a+b \end{vmatrix} = \begin{vmatrix} 1 & a & b+c \\ 0 & b-a & a-b \\ 0 & c-a & a-c \end{vmatrix} = \begin{vmatrix} b-a & a-b \\ c-a & a-c \end{vmatrix} = (b-a)(a-c) - (a-b)(c-a)$

$= ba - bc - a^2 + ac - ac + a^2 + bc - ab = 0$

(4)

$\begin{vmatrix} a+x & a+y & a+z \\ b+x & b+y & b+z \\ c+x & c+y & c+z \end{vmatrix} = \begin{vmatrix} a+x & a+y & a+z \\ 0 & b+y-\dfrac{(a+y)(b+x)}{a+x} & b+z-\dfrac{(a+z)(b+x)}{a+x} \\ 0 & c+y-\dfrac{(a+y)(c+x)}{a+x} & c+z-\dfrac{(a+z)(c+x)}{a+x} \end{vmatrix}$

$= (a+x)\begin{vmatrix} b+y-\dfrac{(a+y)(b+x)}{a+x} & b+z-\dfrac{(a+z)(b+x)}{a+x} \\ c+y-\dfrac{(a+y)(c+x)}{a+x} & c+z-\dfrac{(a+z)(c+x)}{a+x} \end{vmatrix} = (a+x)\begin{vmatrix} \dfrac{(x-y)(b-a)}{a+x} & \dfrac{(x-z)(b-a)}{a+x} \\ \dfrac{(x-y)(c-a)}{a+x} & \dfrac{(x-z)(c-a)}{a+x} \end{vmatrix}$

$= \frac{(b-a)(c-a)}{a+x}\begin{vmatrix} x-y & x-z \\ x-y & x-z \end{vmatrix} = \frac{(b-a)(c-a)}{a+x}\begin{vmatrix} x-y & x-z \\ 0 & 0 \end{vmatrix} = 0$