Question

No sistema matricial dado por {x+y=3A}                                                                                                                             {x-y=2B}  onde A= [2 0] e B= [1 5], os determinantes das matrizes x e y serão, respectivamente :               [0 4]         [3 0]                                                  a) 9, -4 b) -4, 9 c) -3,-9 d) 9,-3 e) -9, -3               

Answer 1

$X + Y = 3*\left[\begin{array}{cc}2&0\\0&4\end{array}\right]$
$X + Y = \left[\begin{array}{cc}6&0\\0&12\end{array}\right]$

$X - Y = 2*\left[\begin{array}{cc}1&5\\3&0\end{array}\right]$
$X - Y = \left[\begin{array}{cc}2&10\\6&0\end{array}\right]$
____________________

$X + Y = \left[\begin{array}{cc}6&0\\0&12\end{array}\right]$
$X - Y = \left[\begin{array}{cc}2&10\\6&0\end{array}\right]$

Somando as equações:

$X+X+Y-Y=\left[\begin{array}{cc}6&0\\0&12\end{array}\right]+\left[\begin{array}{cc}2&10\\6&0\end{array}\right]$
$2X=\left[\begin{array}{cc}8&10\\6&12\end{array}\right]$
$X=\left[\begin{array}{cc}4&5\\3&6\end{array}\right]$


$X + Y = \left[\begin{array}{cc}6&0\\0&12\end{array}\right]$
$\left[\begin{array}{cc}4&5\\3&6\end{array}\right]+Y=\left[\begin{array}{cc}6&0\\0&12\end{array}\right]$
$Y=\left[\begin{array}{cc}6&0\\0&12\end{array}\right]-\left[\begin{array}{cc}4&5\\3&6\end{array}\right]$
$Y=\left[\begin{array}{cc}2&-5\\-3&6\end{array}\right]$
____________________

$det X = det\left[\begin{array}{cc}4&5\\3&6\end{array}\right]=(4*6)-(5*3)=24-15=9$
$detY=det\left[\begin{array}{cc}2&-5\\-3&6\end{array}\right]=(2*6)-([-5]*[-3])=12-15=-3$

Letra D)