Question

4- Resolva o sistema matricial seguinte:

X + Y = A + 4B
X − Y = 3A − 2B

Dado as matrizes:

A = 1 −3 B 2 −1
4 5 4 -7

Answer 1

Dados:

$\boxed{Matriz A=\left[\begin{array}{cc}1&-3\\4&5\end{array}\right] }$

$\boxed{MatrizB=\left[\begin{array}{cc}2&-1\\4&-7\end{array}\right] }$

Resolvendo

$\boxed{\left \{ {{x+y=A+4B} \atop {x-y=3A-2B}} \right. }$

Multiplicando a 2° equação por -1 e somando a primeira temos:

$\boxed{\left \{ {{x+y=A+4B} \atop {(x-y=3A-2B)*-1}} \right. }=\boxed{\left \{ {{x+y=A+4B} \atop {-x+y=-3A+2B}} \right. }=\boxed{2y=-2A+2B}$

Resolvendo para y:

$2y=-2\left[\begin{array}{cc}1&-3\\4&5\end{array}\right] +2\left[\begin{array}{cc}2&-1\\4&-7\end{array}\right]$

$2y=\left[\begin{array}{cc}-2&6\\-8&-10\end{array}\right]+\left[\begin{array}{cc}4&-2\\8&-14\end{array}\right]$

$2y=\left[\begin{array}{cc}2&4\\0&-24\end{array}\right]$

$y=\left[\begin{array}{cc}1&2\\0&-12\end{array}\right]$

Substituindo para encontrar x:

$x+y=A+4B$

$x+\left[\begin{array}{cc}1&2\\0&-12\end{array}\right] = \left[\begin{array}{cc}1&-3\\4&5\end{array}\right]+4\left[\begin{array}{cc}2&-1\\4&-7\end{array}\right]$

$x= \left[\begin{array}{cc}1&-3\\4&5\end{array}\right]+4\left[\begin{array}{cc}2&-1\\4&-7\end{array}\right]-\left[\begin{array}{cc}1&2\\0&-12\end{array}\right]$

$x= \left[\begin{array}{cc}1&-3\\4&5\end{array}\right]+\left[\begin{array}{cc}8&-4\\16&-28\end{array}\right]-\left[\begin{array}{cc}1&2\\0&-12\end{array}\right]$

$x= \left[\begin{array}{cc}8&-9\\20&-11\end{array}\right]$

Resposta: $\boxed{x= \left[\begin{array}{cc}8&-9\\20&-11\end{array}\right]}$ e $\boxed{y=\left[\begin{array}{cc}1&2\\0&-12\end{array}\right]}$