Question

UPF-RS 2014
Dadas as matrizes quadradas A, B e C, de ordem n, e a matriz identidade In de mesma ordem, considere as proposições a seguir, verificando se são verdadeiras (V) ou falsas (F).

( ) (A+B)2 = A2 +2AB+B2
( ) (A-B)2 = A2-B2
( ) C.I=C


A sequência correta de preenchimento dos parênte-
ses, de cima para baixo, é:
a) V-V-V.
b) V-F-V.
c) F-V-V.
d) F-F-V
e) F-F-F

Answer 1

( F ) (A + B)² = A² + 2AB + B²

( F ) (A - B)² = A²-  B²

( V ) C.I = C

Podemos observar conforme acima que a alternativa d é a que corresponde à alternativa correta para esta questão.

Operações com Matrizes

Para resolvermos esta questão vamos imaginar três matrizes quadradas qualquer e a Matriz Identidade igualmente quadrada com mesma quantidade de colunas e linhas que as anteriores:

$A = \left[\begin{array}{cc}1&7&\\3&8&\\\end{array}\right]$         $B = \left[\begin{array}{cc}2&8&\\4&9&\\\end{array}\right]$         $C = \left[\begin{array}{cc}3&9&\\5&10&\\\end{array}\right]$  $I_{n} = \left[\begin{array}{cc}1&0&\\0&1&\\\end{array}\right]$

Vamos verificar se é verdadeira ou falsa a primeira proposição:

$(A+B)^2 = A^2 + 2AB + B^2$  

Somando A + B

$A + B= \left[\begin{array}{cc}1&7&\\3&8&\\\end{array}\right]+ \left[\begin{array}{cc}2&8&\\4&9&\\\end{array}\right]$      $A+B = \left[\begin{array}{cc}1+2&7+8&\\3+4&8+9&\\\end{array}\right]$

$A+B = \left[\begin{array}{cc}3&15&\\7&17&\\\end{array}\right]$

Elevando A + B ao quadrado:

$(A+B)^2 = \left[\begin{array}{cc}3&15&\\7&17&\\\end{array}\right].\left[\begin{array}{cc}3&15&\\7&17&\\\end{array}\right]$   $(A+B)^2 = \left[\begin{array}{cc}3.3+15.7&3.15+15.17&\\7.3+17.7&7.15+17.17&\\\end{array}\right]$

$(A+B)^2 = \left[\begin{array}{cc}9+105&45+255&\\21+119&105+289&\\\end{array}\right]$  $(A+B)^2 = \left[\begin{array}{cc}114&300&\\140&394&\\\end{array}\right]$

Calculando A²:

$A^2 = \left[\begin{array}{cc}1&7&\\3&8&\\\end{array}\right]. \left[\begin{array}{cc}1&7&\\3&8&\\\end{array}\right]$     $A^2 = \left[\begin{array}{cc}1.1+7.3&1.7+7.8&\\3.1+8.3&3.7+8.8&\\\end{array}\right]$

$A^2 = \left[\begin{array}{cc}1+21&7+56&\\3+24&21+64&\\\end{array}\right]$       $A^2 = \left[\begin{array}{cc}22&63&\\27&85&\\\end{array}\right]$  

Multiplicando 2 (AB):

$2.\left[\begin{array}{cc}1&7&\\3&8&\\\end{array}\right].\left[\begin{array}{cc}2&8&\\4&9&\\\end{array}\right]$   $2. \left[\begin{array}{cc}1.2+7.4&1.8+7.9&\\3.2+8.4&3.8+8.9&\\\end{array}\right]$  $2. \left[\begin{array}{cc}2+28&8+63&\\6+32&24+72&\\\end{array}\right]$

$2. \left[\begin{array}{cc}30&71&\\38&96&\\\end{array}\right]$     $2AB=\left[\begin{array}{cc}60&142&\\76&192&\\\end{array}\right]$

Calculando B²:

$B^2 = \left[\begin{array}{cc}2&8&\\4&9&\\\end{array}\right]. \left[\begin{array}{cc}2&8&\\4&9&\\\end{array}\right]$     $B^2 = \left[\begin{array}{cc}2.2+8.4&2.8+8.9&\\4.2+9.4&4.8+9.9&\\\end{array}\right]$$B^2 = \left[\begin{array}{cc}4+32&16+72&\\8+36&32+81&\\\end{array}\right]$     $B^2 = \left[\begin{array}{cc}36&88&\\44&113&\\\end{array}\right]$  

Conferindo a primeira proposição:

$(A+B)^2 = A^2 + 2AB + B^2$

$\left[\begin{array}{cc}114&300&\\140&394&\\\end{array}\right] =\left[\begin{array}{cc}60&142&\\76&192&\\\end{array}\right]+\left[\begin{array}{cc}36&88&\\44&113&\\\end{array}\right]$

$\left[\begin{array}{cc}114&300&\\140&394&\\\end{array}\right] =\left[\begin{array}{cc}60+36&142+88&\\76+44&192+113&\\\end{array}\right]$  

$\left[\begin{array}{cc}114&300&\\140&394&\\\end{array}\right] \cong\left[\begin{array}{cc}96&230&\\120&305&\\\end{array}\right]$  Esta proposição é falsa.      

Vamos verificar se é verdadeira ou falsa a segunda proposição:

(A - B) ² = A² - B²

$A - B= \left[\begin{array}{cc}1&7&\\3&8&\\\end{array}\right]- \left[\begin{array}{cc}2&8&\\4&9&\\\end{array}\right]$     $A - B= \left[\begin{array}{cc}1-2&7-8&\\3-4&8-9&\\\end{array}\right]$

$A - B= \left[\begin{array}{cc}-1&-1&\\-1&-1&\\\end{array}\right]$    $(A - B)^2= \left[\begin{array}{cc}-1&-1&\\-1&-1&\\\end{array}\right].\left[\begin{array}{cc}-1&-1&\\-1&-1&\\\end{array}\right]$

$(A - B)^2= \left[\begin{array}{cc}-1.-1+-1.-1&-1.-1+-1.-1&\\-1.-1+-1.-1&-1.-1+-1.-1&\\\end{array}\right]$

$(A - B)^2= \left[\begin{array}{cc}1+1&1+1&\\1+1&1+1&\\\end{array}\right]$        $(A - B)^2= \left[\begin{array}{cc}2&2&\\2&2&\\\end{array}\right]$

$A^2 - B^2= \left[\begin{array}{cc}22&63&\\27&85&\\\end{array}\right]- \left[\begin{array}{cc}36&88&\\44&113&\\\end{array}\right]= \left[\begin{array}{cc}-14&-25&\\-17&-28&\\\end{array}\right]$

$\left[\begin{array}{cc}2&2&\\2&2&\\\end{array}\right]\cong\left[\begin{array}{cc}-14&-25&\\-17&-28&\\\end{array}\right]$

$(A-B)^2\cong A^2-B^2$

Vamos verificar se e verdadeira ou falsa a terceira proposição:

CI = C

$C = \left[\begin{array}{cc}3&9&\\5&10&\\\end{array}\right].I_{n} = \left[\begin{array}{cc}1&0&\\0&1&\\\end{array}\right]$

$C.In = \left[\begin{array}{cc}3.1+9.0&3.0+9.1&\\5.1+10.0&5.0+10.1&\\\end{array}\right]$

$C.In = \left[\begin{array}{cc}3&9&\\5&10&\\\end{array}\right]$

$C.In = C$

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