Question

Sabendo que a b c
d e f = 6,
g h i
encontre:

a) d e f
g h i
a b c

b) 3a 3b 3c
-d -e -f
4g 4h 4i

c) -3a -3b -3c
d e f
g-4d h-4e i-4f

Answer 1

   $\left[\begin{array}{ccc}a&b&c\\d&e&f\\g&h&i\end{array}\right] =-6 \\ \\ aei+bfg+cdh-ceg-bdi-afh=-6 \\ \\ (aei+bfg+cdh)-(ceg+bdi+afh)=-6 \\ \\ \\ a) \left[\begin{array}{ccc}d&e&f\\g&h&i\\a&b&c\end{array}\right] =cdh+aei+bfg-afh-ceg-bdi= \\ =(aei+bfg+cdh)-(ceg+bdi+afh)=-6 \\ \\ \\ b) \left[\begin{array}{ccc}3a&3b&3c\\-d&-e&-f\\4g&4h&4i\end{array}\right] = \\ =-12aei-12bfg-12cdh+12ceg+12bdi+12af= \\ =-12(aei+bfg+cdh)+12(ceg+bdi+afh)= \\ =-12[(aei+bfg+cdh)-(ceg+bdi+afh)]=-12[-6]=72$

$c) \left[\begin{array}{ccc}-3a&-3b&-3c\\d&e&f\\(g-4d)&(h-4e)&(i-4f)\end{array}\right] = \\ \\ =-3ae(i-4f)-3bf(g-4d)-3cd(h-4e)+3ce(g-4d)+ \\ +3bd(i-4f)+3af(h-4e)=-3aei+12aef-3bfg+12bdf-3cdh+ \\ +12cde+3ceg-12cde+3bdi-12bdf+3afh-12afe= \\ -3(aei+bfg+cdh)+3(ceg+bdi+afh)= \\ =-3[(aei+bfg+cdh)-(ceg+bdi+afh)]=-3[-6]=18$