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{\f1\froman\fcharset2\fprq2 Symbol;}}
{\colortbl;\red0\green0\blue0;\red255\green255\blue255;}
{\*\generator TX_RTF32 10.1.323.500;}
\deftab1134\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080\plain\f0\fs20 PI 9)  Sendo A = cuie:\par a)
(A+B)'\par b)  A'+B'\par c)   (A . B)'\par 6-1\par  B :\par d)  A' \loch\f0\hich\f0 \'95 B'\par e)   B' -A'\par , cai-    ;   P22)   Dadas   as   matrizes   A =
0    0 1      _     I O\par -P \'b0! B=\par Li   oJ\par a)  A . B\par b)  A . C\par P20)  Dada  a  matriz A =\par 3     1 2\par ^   TJ\par ,  encontre\par uma
matriz   n\'e3o-nula   B =\par 4: i]\par tai\par que A \'95 B = C, sendo C uma matriz nula de ordem 2.\par P21)  Calcijie a matriz X nas equa\'e7\'f5es abaixo:\par
[2    3    41     [1     O    -11 ^^^ + i1     o    2j=L5    1        iJ\par b) 2X-\par\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\t
rpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx288\clvertalt\clbrdrr\brdrs\brdrw15\cellx586\clvertalt\clbrdrl\brdrs\brdrw15\clbrdrr\brdrs\brdrw15\cellx845\clvertalt\clbrdrl\brdrs\br
drw15\cellx1123\clvertalt\clbrdrr\brdrs\brdrw15\cellx1440\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080
"1\cell r\cell\cell 3\cell r\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw1
5\cellx288\clvertalt\clbrdrr\brdrs\brdrw15\cellx586\clvertalt\clbrdrl\brdrs\brdrw15\clbrdrr\brdrs\brdrw15\cellx845\clvertalt\clbrdrl\brdrs\brdrw15\cellx1123\clvertalt\clbrdrr\brdrs
\brdrw15\cellx1440\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 2\cell 2\cell =\cell 0\cell 1\cell\intbl\row\pard\tr
owd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx288\clvertalt\clbrdrr\brdrs\brdrw15\cellx586
\clvertalt\clbrdrl\brdrs\brdrw15\clbrdrr\brdrs\brdrw15\cellx845\clvertalt\clbrdrl\brdrs\brdrw15\cellx1123\clvertalt\clbrdrr\brdrs\brdrw15\cellx1440\pard\intbl\tx720\tx1440\tx2160\t
x2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 _3\cell 3-\cell\cell .6\cell 1_\cell\intbl\row\pard\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx504
0\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 c)\par 2    O 2    1\par\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\cl
vertalt\clbrdrl\brdrs\brdrw15\cellx278\clvertalt\clbrdrr\brdrs\brdrw15\cellx614\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360
\tx10080 "1\cell r\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx278
\clvertalt\clbrdrr\brdrs\brdrw15\cellx614\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 1\cell 2\cell\intbl\row\pard\
trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx278\clvertalt\clbrdrr\brdrs\brdrw15\cellx6
14\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 1\cell 3\cell\intbl\row\pard\pard\tx720\tx1440\tx2160\tx2880\tx3600\
tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 P23) Compare os resultados obtidos no exer-cicio anterior e responda: Se A \'95 B = A \'95 C, \'e9 
necess\'e1rio que B = C?\par P24)  Se   A =\par ,   \'e9   poss\'edvel   encontrar\par Bj X2' matriz n\'e3o-nula, tal que A \'95 B = 0? Caso seja poss\'edvel, d\'ea 
um exemplo de B.\par P25)  Calcule;\par a)  X = A=-2A-\'cd-3I,  sabendo  que  A =\par 1    2 3,4\par b)  (A \'95 B)-l-C', sabendoque A =\par B =\par 2   1\par e 
C =\par 5   6 O    1     O\par 0     O    1\par 1       O    O\par Matriz inversa                       ^\par Sabemos que a matriz identidade !\'84 \'e9 da forma:\par
1   o o\par\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx317\clvertalt\cellx634\clverta
lt\clbrdrr\brdrs\brdrw15\cellx960\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 "1\cell 0\cell 0"\cell\intbl\row\pard
\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx317\clvertalt\cellx634\clvertalt\clbrdrr\
brdrs\brdrw15\cellx960\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 0\cell 1\cell 0\cell\intbl\row\pard\trowd\trgaph
40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx317\clvertalt\cellx634\clvertalt\clbrdrr\brdrs\brdrw15
\cellx960\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 _0\cell 0\cell 1.\cell\intbl\row\pard\pard\tx720\tx1440\tx216
0\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 , ...,  !\'84 =\par\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3
\trpaddfr3\trpaddfb3\clvertalt\cellx202\clvertalt\cellx519\clvertalt\cellx836\clvertalt\cellx1450\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx
7920\tx8640\tx9360\tx10080 1\cell 0\cell 0\cell ...    0\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\cl
vertalt\cellx202\clvertalt\cellx519\clvertalt\cellx836\clvertalt\cellx1450\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10
080 0\cell 1\cell 0\cell ...    0\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\cellx202\clvert
alt\cellx519\clvertalt\cellx836\clvertalt\cellx1450\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080
0\cell 0\cell 1\cell ...    0\cell\intbl\row\pard\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 0    0
0\par Considerando  a  matriz  quadrada  A = produtos: \'ae A . I  \'95\par , vamos calcular os seguintes\par AT\par c   d\par 1    O O    1\par a\'951 + b .0 c
. 1+d \'95 O\par \'95 0 + b . li .0 + d \'95 lj\par 63\par\pard\par\tab\tab\par\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx100
80 \'95 I2 \'95 A:\par I2 \'95A =\par c   d\par  01    Ta   b\par\par ^p-a + O-c    l-b + O.d [o -a+l -c   O . b+1 -d\par Assim, A \'95 I2 = I^ . A = A.\par De um
modo geral, para qualquer matriz de ordem n:\par dradal\par A ma,ri. .dent.dade I. \'e9 o elemento neutro da multiplica\'e7\'e3o de matrizes qua-\par Exerc\'edcio
resolvido\par R5) Sendo A =\par  1    4J'\par Solu\'e7\'e3o: A \'95 B =\par \'97s. A       D\par I    O O    1\par B e B . A.\par =>B . A =\par Ent\'e3o:\par Logo:\par
A . B = B . A = I3\par Neste caso, dizemos que A e B s\'e3:\par sao matrizes inversas. 64\par\pard\plain\f0\fs24  \par\tab\tab\par\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5
040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080\plain\f0\fs20 De um modo geral, se A e B s\loch\f0\hich\f0 \'e3o matrizes quadradas de ordem n, dizemos que 
B \'e9 inversa de A se, e somente se:\par A ' B = B . A = i.\par Se B \'e9 inversa de A, ent\'e3o B = A  ^ onde A  ^ \'e9 matriz inversa de A.\par Dizemos, ent\'e3o, 
que:\par ~^ \'e9 inversa de A se, e somente se:\par \'95 uma matriz quadrada A \'e9 invert\'edvel se existe A  *; e uma matriz quadrada A \'e9 singular se n\'e3o 
existe A\par Exerc\'edcio resolvido\par R6) Calcular A  \\ sabendo que:\par b) B =\par\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\tr
paddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx298\clvertalt\cellx586\clvertalt\clbrdrr\brdrs\brdrw15\cellx903\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx648
0\tx7200\tx7920\tx8640\tx9360\tx10080 1\cell 1\cell 0\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clver
talt\clbrdrl\brdrs\brdrw15\cellx298\clvertalt\cellx586\clvertalt\clbrdrr\brdrs\brdrw15\cellx903\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx79
20\tx8640\tx9360\tx10080 1\cell 0\cell 1\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\
brdrs\brdrw15\cellx298\clvertalt\cellx586\clvertalt\clbrdrr\brdrs\brdrw15\cellx903\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9
360\tx10080 2\cell 1\cell 1\cell\intbl\row\pard\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 Solu\'e7\'e3o:\par 
a) A =\par 3   2 1    1\par Fazendo A  ^ = p   ^   (se existir) e aplicando a rela\'e7\'e3o A \'95 A  ^ = !\'84, obtemos:\par 3x + 2z x+z         y+t\par  O'\par 
0    1\par Pela igualdade de matrizes, vem:\par  t=l\par  _2 e t = 3\par Logo:\par 65\par\pard\plain\f0\fs24  \par\tab\tab\par\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\
tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080\plain\f0\fs20 Para verificarmos se o resultado obtido est\loch\f0\hich\f0 \'e1 correto, devemos efetuar o produto 
A \'95 A" e obter a identidade I, :\par 3    2 1    1.\par 1    -2 1        3\par 3 \'95 1+2 \'95 (-1)    3 . (-2) + 2 \'95 3 1 . 1 + 1 \'95(-!)    1 \'95(-2)+l 
\'95 3\par 1    O O\par Neste caso, A e A   ' s\'e3o matrizes inversas.\par b) A =\par\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\tr
paddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx298\clvertalt\cellx586\clvertalt\clbrdrr\brdrs\brdrw15\cellx903\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx648
0\tx7200\tx7920\tx8640\tx9360\tx10080 "1\cell 1\cell\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvert
alt\clbrdrl\brdrs\brdrw15\cellx298\clvertalt\cellx586\clvertalt\clbrdrr\brdrs\brdrw15\cellx903\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx792
0\tx8640\tx9360\tx10080 1\cell 0\cell 1\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\b
rdrs\brdrw15\cellx298\clvertalt\cellx586\clvertalt\clbrdrr\brdrs\brdrw15\cellx903\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx93
60\tx10080 2\cell 1\cell\cell\intbl\row\pard\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 Fazendo A   ' 
=\par abe\par d    e    f g    h    i\par (se existir) e aplicando a rela\'e7\'e3o A \'95 A   ' = !\'84, obtemos:\par\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\t
rpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx288\clvertalt\cellx586\clvertalt\clbrdrr\brdrs\brdrw15\cellx874\clvertalt\clbrdrl\brdrs\brdrw15\clbrdrr\
brdrs\brdrw15\cellx1085\clvertalt\clbrdrl\brdrs\brdrw15\cellx1383\clvertalt\cellx1681\clvertalt\clbrdrr\brdrs\brdrw15\cellx1988\clvertalt\clbrdrl\brdrs\brdrw15\clbrdrr\brdrs\brdrw1
5\cellx2190\clvertalt\clbrdrl\brdrs\brdrw15\cellx2488\clvertalt\cellx2766\clvertalt\clbrdrr\brdrs\brdrw15\cellx3083\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx576
0\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 1\cell 1\cell 0"\cell\cell a\cell b\cell c\cell\cell '1\cell 0\cell 0"\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\tr
paddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx288\clvertalt\cellx586\clvertalt\clbrdrr\brdrs\brdrw15\cellx874\clvertalt\clbrdrl\brdr
s\brdrw15\clbrdrr\brdrs\brdrw15\cellx1085\clvertalt\clbrdrl\brdrs\brdrw15\cellx1383\clvertalt\cellx1681\clvertalt\clbrdrr\brdrs\brdrw15\cellx1988\clvertalt\clbrdrl\brdrs\brdrw15\cl
brdrr\brdrs\brdrw15\cellx2190\clvertalt\clbrdrl\brdrs\brdrw15\cellx2488\clvertalt\cellx2766\clvertalt\clbrdrr\brdrs\brdrw15\cellx3083\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\t
x4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 1\cell 0\cell 1\cell \'95\cell d\cell e\cell f\cell =\cell 0\cell 1\cell 0\cell\intbl\row\pard\trowd\trgaph40\trleft0
\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx288\clvertalt\cellx586\clvertalt\clbrdrr\brdrs\brdrw15\cellx874\
clvertalt\clbrdrl\brdrs\brdrw15\clbrdrr\brdrs\brdrw15\cellx1085\clvertalt\clbrdrl\brdrs\brdrw15\cellx1383\clvertalt\cellx1681\clvertalt\clbrdrr\brdrs\brdrw15\cellx1988\clvertalt\cl
brdrl\brdrs\brdrw15\clbrdrr\brdrs\brdrw15\cellx2190\clvertalt\clbrdrl\brdrs\brdrw15\cellx2488\clvertalt\cellx2766\clvertalt\clbrdrr\brdrs\brdrw15\cellx3083\pard\intbl\tx720\tx1440\
tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 2\cell 1\cell 1_\cell\cell -g\cell h\cell i ^\cell\cell _0\cell 0\cell L\cell\intbl\row\pard\pa
rd\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 a+d             b+e            c+f\par a+g             b+h  
c+i\par _2a + d + g     2b + e + h     2c + f +\par Pela igualdade de matrizes, vem:\par\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\
trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\clbrdrr\brdrs\brdrw15\cellx250\clvertalt\clbrdrl\brdrs\brdrw15\cellx538\clvertalt\cellx826\clvertalt\clbrdrr\brdrs\brdrw15\cellx1152\pard\
intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080\cell "1\cell 0\cell 0"\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\
trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\clbrdrr\brdrs\brdrw15\cellx250\clvertalt\clbrdrl\brdrs\brdrw15\cellx538\clvertal
t\cellx826\clvertalt\clbrdrr\brdrs\brdrw15\cellx1152\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 
=\cell 0\cell 1\cell 0\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\clbr
drr\brdrs\brdrw15\cellx250\clvertalt\clbrdrl\brdrs\brdrw15\cellx538\clvertalt\cellx826\clvertalt\clbrdrr\brdrs\brdrw15\cellx1152\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320
\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080\cell 0\cell 0\cell 1\cell\intbl\row\pard\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\
tx8640\tx9360\tx10080  =0 (absurdo)\par  = O (absurdo)\par J c + i = O^i= -c L2c+f+i=1^2c-c-c=1^0=l (absurdo)\par Ent\'e3o, encontramos tr\'eas sistemas imposs\'edveis. 
Logo, n\'e3o existe a matriz A~i.\par Propriedades da matriz inversa\par Sendo A e B matrizes quadradas de ordem n e\par inversas, s\'e3o v\'e1lidas as seguintes 
propriedades:\par 2) (A\par 3)\par 66\par  respectivamente suas\par\pard\plain\f0\fs24  \par\tab\tab\par\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\t
x7920\tx8640\tx9360\tx10080\plain\f0\fs20 Exerc\loch\f0\hich\f0 \'edcios resolvidos\par R7) Sendo A, B e X matrizes invertiveis de ordem n, isolar X em (X \'95 A)\par 
Solu\'e7\'e3o:\par Aplicando uma das propriedades da matriz inversa, obtemos;\par (X . A)-i = B^A-i \'95 X-i = B\par Multiplicando os dois membros \'e0 esquerda 
por A, encontramos:\par A . A"' \'95 X-i =A \'95 B\par Mas A \'95 A"' = I^. Ent\'e3o:\par In \'95 X-i=A . B\par Como In \'e9 elemento neutro na multiplica\'e7\'e3o 
de matrizes, temos:\par X-'=A \'95 B\par Elevando os dois membros a  -1, obtemos:\par  = B.\par R8) Calcular X em 2 \'95             + X' =              , sabendo 
que X \'e9 uma matriz quadrada de\par \\       ordem 2.\par\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\cellx672\
clvertalt\cellx1219\clvertalt\cellx1670\clvertalt\cellx2006\clvertalt\cellx2630\clvertalt\cellx3206\clvertalt\cellx3436\clvertalt\cellx3734\pard\intbl\tx720\tx1440\tx2160\tx2880\tx
3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 Solu\'e7\'e3o\cell\cell\cell\cell\cell\cell\cell\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\
trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\cellx1219\clvertalt\cellx2630\clvertalt\cellx3206\clvertalt\cellx3734\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3
600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 Multiplicando\cell 0 escalar 2 pela\cell matriz\cell , vem\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\
trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrr\brdrs\brdrw15\cellx672\clvertalt\clbrdrl\brdrs\brdrw15\cellx1219\clvertalt\cellx1670\clvertalt
\cellx2630\clvertalt\cellx3206\clvertalt\cellx3734\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 L4 
10_\cell\cell\cell loJ\cell\cell\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\cellx1670\clvert
alt\cellx2630\clvertalt\cellx3206\clvertalt\cellx3734\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 
Isolando a matriz X\cell ', temos:\cell\cell\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\cell
x672\clvertalt\cellx1219\clvertalt\cellx1670\clvertalt\clbrdrr\brdrs\brdrw15\cellx2006\clvertalt\clbrdrl\brdrs\brdrw15\cellx2630\clvertalt\cellx3206\clvertalt\clbrdrr\brdrs\brdrw15
\cellx3436\clvertalt\clbrdrl\brdrs\brdrw15\cellx3734\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 
-G\cell \'edoJ"\cell\cell 10\cell\cell 1-2\cell 3' 0\cell\cell\intbl\row\pard\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080
 Calculando a transposta de ambos os membros, obtemos:\par -2    0\par  3        O\par L\'f4   o   0^\par R9) Sendo A = A \'95 X-B = C. Solu\'e7\'e3o:\par Isolando 
a matriz X, temos:\par 1    21           [8    91\par e C =            , calcular a matriz X de ordem 2, tal que\par 3    4J.             L4    \'f4J\par (C + B) 
=\par (C + B)\par 67\par\pard\plain\f0\fs24  \par\tab\tab\par\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080\plain\f0\fs20\p
ar C\loch\f0\hich\f0 \'e1lculo de A\par ijuir\par c         ,     ._,     Ta    bl\par hazendo A  ^ = I       ^   (se existir) e aplicando a rela\'e7\'e3o A~i \'95 
A = !\'84, vem:\par ri 01\par Lo ij\par a   b1    |~2    li    n    01    r2a + b    a ^1    lj    |_0    lj    |_2c + d    c Pela igualdade de matrizes, vem:\par 
^, u   ~A  ^a=l e b= -1 a + b = O\par c'Vdl?=^'^=-\'ed^d = 2 Ent\'e3o:\par _r 1  -1\par "Ul       2 C\'e1lculo de X:\par X = A"i \'95 (C + E\par -1\par  7  lo\par 
5    9\par Exerc\'edcios prapostDS\par P26)  Calcule as  inversas  das seguintes  matrizes:\par c)  C =\par 3    4\par d)  D =\par e)   E =\par Ll_\par 1     0  
0\par 2    1\par 1     1\par 0     O     O\par "2     1     1'\par 1     O     O 3     1     1\par P27) Dadas as matrizes abaixo, identifique quais s\'e3o invertiveis 
e quais s\'e3o singulares:\par  0]\par 2    2    2 .3    2    0_\par P28)  Sendo A, B e X matrizes invert\'edveis de ordem n, isole X nas equa\'e7\'f5es abaixo:\par 
a)   A . X= B\par b)   A . X . 8 = B\par c)   (A . X)--! =A\par d)   (A . X)  . B-' =A . B\par e)   X \'95 (A . B)--' =A--'\par f)   X . A=B P29)   Calcule a  matriz:\par 
a)   B^xa,   sabendo   que   A\par  B =\par P    '\par L2    1\par b) X,\par que\par 2\par -1\par ro  11\par L2    oJ\par 68\par II\par\pard\plain\f0\fs24  \par\tab\tab\par\pard\tx
720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080\plain\f0\fs20 c)   X, sabendo que X \loch\f0\hich\f0 \'95A=B,\par 
n     21          [2    31\par Lo  ij        Li   2j\par d)  X, sabendo que(A + X)' = = 2 . (A+B) . C,\par -[\'e0 4] 5-[\'ed 2\par e)  A, sabendo que X \'95A=lo 
e X =\par f)  X, sabendo que 2X-\par\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx288\c
lvertalt\clbrdrr\brdrs\brdrw15\cellx605\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 1\cell 0\cell\intbl\row\pard\tr
owd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx288\clvertalt\clbrdrr\brdrs\brdrw15\cellx605
\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 0\cell 1\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0
\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx288\clvertalt\clbrdrr\brdrs\brdrw15\cellx605\pard\intbl\tx720\tx1440\tx2160\tx2880\
tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 .1\cell 0\cell\intbl\row\pard\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx792
0\tx8640\tx9360\tx10080 n   O  ii                n   01\par 4o   1   oj-  \'b0   ^   Ho   oj\par 1    o\par I   P31)   Resolva   as   equa\'e7\'f5es   abaixo,   
sabendo que X \'e9 uma matriz quadrada de ordem 2:\par d)\par 3 1 -5 7\par P32)   Prove  as  senten\'e7as  abaixo, propriedades:\par b)\par  usando  as  -B-B\par 
P30)   Calcule;\par "1     2 3    4 6\par 3    2\par -2\par\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\b
rdrs\brdrw15\cellx288\clvertalt\cellx595\clvertalt\clbrdrr\brdrs\brdrw15\cellx883\clvertalt\clbrdrl\brdrs\brdrw15\cellx1152\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx50
40\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 "2\cell 0\cell r\cell t\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpaddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\tr
paddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx288\clvertalt\cellx595\clvertalt\clbrdrr\brdrs\brdrw15\cellx883\clvertalt\clbrdrl\brdrs\brdrw15\clbrdrr\brdrs\brdrw15\cellx11
52\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 0\cell 2\cell 1\cell\cell\intbl\row\pard\trowd\trgaph40\trleft0\trpa
ddl40\trpaddt0\trpaddr40\trpaddb0\trpaddfl3\trpaddft3\trpaddfr3\trpaddfb3\clvertalt\clbrdrl\brdrs\brdrw15\cellx288\clvertalt\cellx595\clvertalt\clbrdrr\brdrs\brdrw15\cellx883\clver
talt\clbrdrl\brdrs\brdrw15\cellx1152\pard\intbl\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080 1\cell 2\cell 0.\cell\cell\intbl\r
ow\pard\pard\tx720\tx1440\tx2160\tx2880\tx3600\tx4320\tx5040\tx5760\tx6480\tx7200\tx7920\tx8640\tx9360\tx10080  comut\'e1yeis\par Dizemos que as matrizes A e B s\'e3o 
comut\'e1veis se, e somente se:\par A . B = B . A\par Exerc\'edcios resolvidcs\par RIO) Dar tr\'eas exemplos de matrizes comut\'e1veis com A =\par 1    2 3    O\par 
Solu\'e7\'e3o:\par X    y\par Suponhamos que B = | '   "' | seja comut\'e1vel com A. Ent\'e3o:\par 1    2 3   O\par 69\par\pard\plain\f0\fs24  \par }
