Question

Sa se calculeze valoarea determinantului
$\left[\begin{array}{ccc}-2a&a+b&a+c\\b+a&-2b&b+c\\c+a&c+b&-2c\end{array}\right]$

Answer 1

$\left|\begin{array}{ccc}-2a&a+b&a+c\\b+a&-2b&b+c\\c+a&c+b&-2c\end{array}\right| \overset{\begin{array}{lcl}C_1\to C_1+C_2\\C_3\to C_3+C_2\end{array}}{=\joinrel=} \\ \\\\ =\left|\begin{array}{ccc}b-a&a+b&2a+b+c\\a-b&-2b&c-b\\2c+a+b&c+b&b-c\end{array}\right| \overset{\begin{array}{lcl}L_1\to L_1+L_2\\L_3\to L_3+L_2\end{array}}{=\joinrel=}$

$=\left|\begin{array}{ccc}0&a-b&2a+2c\\a-b&-2b&c-b\\2c+2a&c-b&0\end{array}\right| =\\\\ \\ = 2(a-b)(c-b)(a+c) + 2(a-b)(c-b)(c+a)+8b(a+c)^2 = \\ \\ = 4(a-b)(c-b)(a+c)+8b(a+c)^2 = \\ \\ = 4(a+c)\Big[(a-b)(c-b)+2b(a+c)\Big]$

La o formă mai simplă de atât nu o pot aduce.