Question

se A= [2 7 -1 4] e B=[3 -2 6 0] determine a matriz x em cada caso:

a) x+b=a
b) x+b=2a
c) 2a+x=3b

Answer 1

Olá



Matrizes, Equação Matricial.


$\displaystyle \mathsf{A= \left[\begin{array}{ccc}2&7\\-1&4\\\end{array}\right] \qquad\qquad\qquad\qquad B= \left[\begin{array}{ccc}3&-2\\6&0\\\end{array}\right] }$



A)

x + b = a

x = a - b



$\displaystyle \mathsf{X=\left[\begin{array}{ccc}2&7\\-1&4\\\end{array}\right] - \left[\begin{array}{ccc}3&-2\\6&0\\\end{array}\right] }\\\\\\\\\mathsf{X=\left[\begin{array}{ccc}(2-3)&(7-(-2))\\(-1-6)&(4-0)\\\end{array}\right] }\\\\\\\\\boxed{\mathsf{ X=\left[\begin{array}{ccc}-1&9\\-7&4\\\end{array}\right] }}$




B)

X + B = 2A

X = 2A - B



$\mathsf{2A~=~2\cdot\left[\begin{array}{ccc}2&7\\-1&4\\\end{array}\right]}\\\\\\\\\mathsf{2A= \left[\begin{array}{ccc}4&14\\-2&8\\\end{array}\right]}$


$\mathsf{X=\mathsf{ \left[\begin{array}{ccc}4&14\\-2&8\\\end{array}\right]}-\left[\begin{array}{ccc}3&-2\\6&0\\\end{array}\right] }}\\\\\\\\\mathsf{X=\left[\begin{array}{ccc}(4-3)&(14-(-2))\\(-2-6)&(8-0)\\\end{array}\right] }}\\\\\\\\\boxed{\mathsf{X=\left[\begin{array}{ccc}1&16\\-8&8\\\end{array}\right] }}}$





C)

2A + X = 3B

X = 3B - 2A



$\mathsf{3B=3\cdot \left[\begin{array}{ccc}3&-2\\6&0\\\end{array}\right] }}\\\\\\\\\mathsf{3B=\left[\begin{array}{ccc}9&-6\\18&0\\\end{array}\right] }}\\\\\\\\2A= \left[\begin{array}{ccc}4&14\\-2&8\\\end{array}\right]\qquad\qquad\qquad\Longleftarrow\text{(obtido no item B)}}$



$\mathsf{X=\left[\begin{array}{ccc}9&-6\\18&0\\\end{array}\right] - \left[\begin{array}{ccc}4&14\\-2&8\\\end{array}\right]}\\\\\\\\\mathsf{X= \left[\begin{array}{ccc}(9-4)&(-6-14)\\(18-(-2))&(0-8)\\\end{array}\right]}\\\\\\\\\boxed{\mathsf{X= \left[\begin{array}{ccc}5&-20\\20&-8\\\end{array}\right]}}$