Question

24. Dadas as matrizes A= (4 -1) (2 0) e B= (2 0) (3 -1) calcule:

25. Considere as matrizes a seguir..

Answer 1

Obs.: Como são questões que levam tempo para resolver de forma adequada aqui na plataforma, resolvi detalhadamente as questões da 24 e na 25 coloquei apenas o gabarito. Caso fique com duvidas em alguma, por favor volte a posta-las individuamente. Obrigado.

24.

Vamos começar determinando as matrizes Identidade (presente na letra b) e a matriz nula 2x2 (presente na letra c).

$I_2~=~\left[\begin{array}{ccc}1&0\\0&1\end{array}\right]\\\\\\0_{2x2}~=~\left[\begin{array}{ccc}0&0\\0&0\end{array}\right]$

a)

$B-A~=~\left[\begin{array}{ccc}2&0\\3&-1\end{array}\right]~-~\left[\begin{array}{ccc}4&-1\\2&0\end{array}\right]\\\\\\\\B-A~=~\left[\begin{array}{ccc}2-4&0-(-1)\\3-2&-1-0\end{array}\right]\\\\\\\\B-A~=~\left[\begin{array}{ccc}-2&1\\1&-1\end{array}\right]$

b)

$A-(B+I_2)~=~\left[\begin{array}{ccc}4&-1\\2&0\end{array}\right]-\left(\left[\begin{array}{ccc}2&0\\3&-1\end{array}\right]+\left[\begin{array}{ccc}1&0\\0&1\end{array}\right]\right)\\\\\\\\A-(B+I_2)~=~\left[\begin{array}{ccc}4&-1\\2&0\end{array}\right]-\left[\begin{array}{ccc}2+1&0+0\\3+0&-1+1\end{array}\right]\\\\\\\\A-(B+I_2)~=~\left[\begin{array}{ccc}4&-1\\2&0\end{array}\right]-\left[\begin{array}{ccc}3&0\\3&0\end{array}\right]$

$A-(B+I_2)~=~\left[\begin{array}{ccc}4-3&-1-0\\2-3&0-0\end{array}\right]\\\\\\\\A-(B+I_2)~=~\left[\begin{array}{ccc}1&-1\\-1&0\end{array}\right]$

c)

$B~-~(A+0_{2x2})~=~\left[\begin{array}{ccc}2&0\\3&-1\end{array}\right]-\left(\left[\begin{array}{ccc}4&-1\\2&0\end{array}\right]+\left[\begin{array}{ccc}0&0\\0&0\end{array}\right]\right)\\\\\\\\B~-~(A+0_{2x2})~=~\left[\begin{array}{ccc}2&0\\3&-1\end{array}\right]-\left[\begin{array}{ccc}4+0&-1+0\\2+0&0+0\end{array}\right]$

$B~-~(A+0_{2x2})~=~\left[\begin{array}{ccc}2&0\\3&-1\end{array}\right]~-~\left[\begin{array}{ccc}4&-1\\2&0\end{array}\right]\\\\\\\\B~-~(A+0_{2x2})~=~\left[\begin{array}{ccc}2-4&0-(-1)\\3-2&-1-0\end{array}\right]\\\\\\\\B~-~(A+0_{2x2})~=~\left[\begin{array}{ccc}-2&1\\1&-1\end{array}\right]$

25.

Para determinar a matriz A transposta (utilizada nas letras "e" e "f") trocamos linha por coluna e ficamos com:

$A^t~=~\left[\begin{array}{ccc}2&1\\0&-5\end{array}\right]$

a)

$A+B~=~\left[\begin{array}{ccc}8&-7\\10&-4\end{array}\right]$

b)

$B+C~=~\left[\begin{array}{ccc}9&-9\\10&-7\end{array}\right]$

c)

$B-A+C~=~\left[\begin{array}{ccc}7&-9\\9&-2\end{array}\right]$

d)

$C-B+A~=~\left[\begin{array}{ccc}-1&5\\-7&-14\end{array}\right]$

e)

$A+A^t~=~\left[\begin{array}{ccc}4&1\\1&-10\end{array}\right]$

f)

$A-A^t~=~\left[\begin{array}{ccc}0&-1\\1&0\end{array}\right]$