Question

1) Calcule a transposta e a inversa das seguintes matrizes

a) A= |1 2|
|0 -6|

b) B= |2 8|
|3 4|

c) C= |2 -4|
|3 -4|

d) D= |1 -1|
|-2 3|




Agradeço quem puder ajudar! :)

Answer 1

1)a)
$\left[\begin{array}{ccc}x&y\\z&a\end{array}\right]$
Ficará
$\left[\begin{array}{ccc}x&z\\y&a\end{array}\right]$

R: $\left[\begin{array}{ccc}1&0\\2&-6\end{array}\right]$

$\left[\begin{array}{ccc}1&2\\0&-6\end{array}\right]$.$\left[\begin{array}{ccc}a&b\\c&d\end{array}\right]$=$\left[\begin{array}{ccc}a+0&b+0\\2a-6c&2b-6d\end{array}\right]$=$\left[\begin{array}{ccc}1&0\\0&1\end{array}\right]$$\left \{ {{a=1} \atop {2a-6c=0}} \right. \left \{ {{b=0} \atop {2b-d=1}} \right.$
a=1
2(1)-6c=0 ⇒2-6c=0 ⇒-6c=-2⇒c=-2/-6⇒c=2/6
____
b=0
2x0-d=1 ⇒ -d=1 ⇒ d=-1

R: $\left[\begin{array}{ccc}1&0\\2/6&-1\end{array}\right]$

b)R: $\left[\begin{array}{ccc}2&3\\8&4\end{array}\right]$

Repetindo o mesmo processo do a) chegaremos aos sistemas:

$\left \{ {{2a+8c=1} \atop {3a+4c=0}} \right.$ $\left \{ {{2b+8d=0} \atop {3b+4d=1}} \right.$
Resolvendo teremos: a=-1/4 c=3/16
e b=1/2 d=-1/8

R: $\left[\begin{array}{ccc}-1/4&1/2\\3/16&-1/8\end{array}\right]$

c)R: $\left[\begin{array}{ccc}2&3\\-4&-4\end{array}\right]$

$\left \{ {{2a-4c=1} \atop {3a-4c=0}} \right.$ $\left \{ {{2b-4d=0} \atop {3b-4d=1}} \right.$
Resolvendo teremos: a=-1 c=-3/4 e b=1 d=1/2

R: $\left[\begin{array}{ccc}-1&1\\-3/4&1/2\end{array}\right]$

d)R: $\left[\begin{array}{ccc}1&-2\\-1&3\end{array}\right]$

$\left \{ {{a-c=1} \atop {-2a+3c=0}} \right.$ $\left \{ {{b-d=0} \atop {-2b+3d=1}} \right.$

Resolvendo teremos: c=2 a=3 e b=1 d=1

R: $\left[\begin{array}{ccc}3&1\\2&1\end{array}\right]$

As respostas estão na ordem do enunciado. =)