Lista Exerci1 Gabarito

a

A  =

�

A+B DA

1 2 2 1

−

3 1

AC  DB

�

B=

� −

BC  A

2 0 1 3 0 1

�

C  =

−

1 2 4 5 1 0

A

6 1 BC 2×1

C D  = D1×2 DA  = A2×3 DB = B2×3

2 4 8

=

2

−1

�

.

B

A

C 

A2×3 C 3×1

B

C 

B2×3 C 3×1

1 2 4

C 

D

C 3x1

A

D1×2

B

D1×2

C D3x2

0 3 6 DA1×3

D

7 0 1 DB1×3

D A  =

−A

�

D  =

−

15 4 AC 2×1

BC  =

1 2 4

CD D.

� − � � � − � � −   −−  � � � − �

A+B =

AC  =

−   

� −

1 2

−2 −3 − −1 1

�

−D

=

�

1 2 2 1

−

3 1

�

� − � 2 1

−1

c23

  − −

1 5 4 1

−2 −2

4 2

�   −−

5 1 5 2 5 2

−4

2

 �    =

c11 c21 c31 c41

c12 c22 c32 c42

c13 c23 c33 c43

c14 c24 c34 c44

 

.

c23 c23  = 21

A  = B 2 =  A B

B

B=

B B 2 =  A

a2 =  d 2

�

1

3

�

B2 =

, B 2 =  BB

A

� �× � � � a b c d

a b c d

a  =

±d 2bd =

−2 2cd = −4

a  =  d

−

−2

 =

−d

bd  =

−1 cd = −2

d2 + b(2b) = 3

 1 d2

)=3 d2

d4 + 2 = 3d2 d2 =  y y2 y  = 1

y  = 2 d =

√  d = 2

√  a  = 2

−1 b  = √  2

±

− 3y + 2 = 0 √  2

−2 c  = √  2

d  =

±1

� √  � √  � −√  � −√  � − � − � −� −1 √ 

2 −2 √ 

B=

2

2

2

√  d = − 2 d =

−1

√  a  = − 2

a  =

−1

d = 1 a  = 1 b  =

b  = √ 2

b  = 1

−1

1

2

c  = √ 2

2 √ 2

B=

2

c  = 2 c  =

−2

B= B=

1 2

1 1

1 2

1 1

−

−

−

1

d

c  = 2b

d2 + 2b2 = 3

d2 + 2(

ab + bd  = 2 a  =  d

b  =

c  = 2b

d

a2 + bc ab + bd ac + cd bc + d2

a2 + bc = 3 bc +  d2 = 3

a  =  d

a  =  d b  =  b  =

−

a b c d

−db + bd = 0 = −2 −2 ac + cd  = −4

3 4

� �

a  =

ab + bd  =

�

1 √ 

2

2

�

x2 = 1

x=1 I 

2

X  =  I 

x  =

 −1

2 2

×2

0 1 A 2

2

×2

A =  A

�

A  =

2x

At =  A

x

A  =  A t

2

x2

−1

0

At

�

.

A

A  =

�

2x

� �

x2

2

−1

2

 =

0

2x

2

−1

0

x

�

 =  A t . Atij

Aij x2 = 2x

− 1.

2

= 0,

(x

− 1)

x  = 1

A  =

 

1 2 4

−3 2 1 −3 −3 −1

AB =  AC 

AB =

 

− −

3 1 3

B=

 

−3

0 15 0 15 0

AB =  AC 

A  =

 −

2 1 1

−3 −5 4 5 −3 −4

1 2 1

−

4 1 0 1 1 1 2 1 2

 −  − 1 5 5

 

AC  =

B =  C 

 

B=

− −

1 1 1

C  =

3 3 3

− −

5 5 5

− −

 

3 1 3

 

−3

2 3 2

0 15 0 15 0

C  =

1 2 5

−1 −2 − −1 −1 − −1 0

 −  − 1 5 5

 −

2 1 1

 

,

,

−2 −4 3 4 −2 −3

 

.

×2

AB =  BA  = 0 AC  =  A

CA  =  C  ACB =  C BA A2

(A

2

± B)

2

=  A + B

2

 

AB =

AC  =

 −

2 1 1

0 0 0 0 0 0 0 0 0

−3 −5 4 5 −3 −4

ABC  = AB

   

BA  =

CA  =

AC  = A

−B

2

= (A +  B )(A

     − − − − − 

− B)

0 0 0 0 0 0 0 0 0 2 1 1

2 3 2

4 4 3

AB = 0

CBA =

CAB =  AB = 0

(A + B )(A

− B) = A

2

+ AB

− BA − B

2

(A + B )(A (A

± B)

2

=  A 2

± 2AB + B

2

AB =  BA 2

AB =  BA  = 0

(A

2

± B)

=  A 2 + B 2 .

AB = AC 

A,B,C 

2

− B) = A − B .

Y 

Y A = I 

B =  C 

B =  I B = (Y A)B =  Y  (AB ) =  Y  (AC ) = (Y A)C  =  I C  =  C 

A  =

� − � 2 1 3 2

A2 =

(A + B )t =  B t + At. AB = 0

A  = 0

AB = 0

BA  = 0

� � 4 1 9 4

B=0 AA

( A)( B ) =

− −

A

B

−(AB).

A  = 0

A BA

I 

� − �� − � � � 2 1 3 2

A2 = AA =

2 1 3 2

7 0 0 7

=

A2

A

At + B t

A AB

B A1×2 = 0

A  =

�

1 2 2 1

3 1

−

 

E  =

1 2 4

�

−3 2 1 −3 −3 −1

A  =

B=

� −

B=

2 0 1 3 0 1

� �

� −

 

1 2 2 1

�

C  =

1 5 4 1

−2 −2

4 2

 

�

1

2

2

1

3

1

4

3 7

1 0 0 1

−

1

 

1

1

�

1 0 2 3 0 1

1 0 0 0

2

�

2

0

 

2

 →

L2

D  =

2 3 2

�

1 L2 8

�

1

� −

5 1 5 2 5 2

 → − 13 L

L2

3 7

2

×2

�

 

−1 −2 − −1 −1 − −1 0

5

−4

−

1 2 5

3

 → L − 3L

1

−2 8 −4

− −

L2

 → L − 2L

L1

   

G  =

1 0

 → L − 2L

L1

� �

4 2

4 1 0 1 1 1 2 1 2

2

�

1 5 4 1

C  =

1 2 1

1 0 2 0 0 1

 → L − 5L  → L  + 4L  → L  + L

L2 L3 L4

2 3

�

L2

 → L  + L

�   − −

3 1

 

1 2 3 0 1 73

L1

 → L − 2L

�

2 0 1 3 0 1

F  =

−

−2 −2

−

  − −  

3

B

2

�

� �

1 0 0 0

−

2 5

�

 → − 15 L

L2

2

1 0 0 0 0 1

 

1 0 0 0

−2 1 −4 0

 

 → L  + 2L L  → L  + 4 L L1

1

2

3

3

2

 

1 0 0 0

0 1 0 0

 

 −     A  =

E  =

2 6 6 2 7 6 2 7 7

1 0 2 1 0 2 1 3 0

F  =

   

 

1 2 4 0 2 1 3 1 2

0 0 5 5

0 0 0 7

 

2x

B=

1 1 1 1

0 3 3 3

 

 

 =

G =

 

� � 1 4 2 7

C  =

0 0 0 k1 0 0 k2 0 0 k3 0 0 k4 0 0 0

D  =

 

H  =

− y + 3z  4x − 3y + 2 z 

= 11 = 0 x + y + z  = 6 3x + y + z  = 4 AX  =  b

[A b]

|

A  =

 

2 4 1 3

−1 −3

3 2 1 1 1 1

 

2 4 1 3

 

X  =

    x y z 

b  =

3 2 1 1 1 1

     ·    

 

−1 −3

−1 −3

x y z 

 =

[ A b]

|

2 4 1 3

3 11 2 0 1 1 6 1 1 4

 

11 0 6 4

 

� − �    

11 0 6 4

 

3 6 4 5

k 0 0 0 1 k 0 0 0 1 k 0 0 0 1 k

 

2 4 1 3



−1 −3

3 11 2 0 1 1 6 1 1 4

 

1 0 0 0

1 1 3 7

 

− − −

1 1 1 2



 

1 4 2 3

  −  − 6 7 1 24

− −

1 3 1 1



1 6 2 0 3 11 1 4

 

1 0 0 0

0 1 0 0

 

1 0 0 0

0 1 0 0

 



0 1 4 5

−1

7 20 25

1 0 0 0

1 7 3 2

− − − − −

 



 

1 2 1 2

1 0 0 0

 −  −  − 6 24 1 14

0 1 0 0

0 1 1 5

−1

7 5 25



 

  

1 0 0 0

 

1 7 3 1

1 2 1 1

− − − 1 0 0 0

0 1 0 0

0 0 1 0

x y z 

x = y = z  =

 =

−1

2 5 x=

 −1

3x + 5 y = 1 2x + z  = 3 5x + y z  = 0

−

 − 

4x + 3 y = 2 5x 4y = 0 2x y = k

− 

4 5 2

3 2 4 0 1 k

− −

    

1 5 2

−1 −4 −1

2 0 k

− −

    

1 0 0

−1

2 1 10 1 k 4

− −

 � �  −  ·  −  1 0 0 1 0 0

1 2 5 0

1 2 5 0

z  = 5

 

 −  −  −   6 24 1 7

   −   ·    

0 0 1 0

  y  = 2

 

x y

 =

8 10 k + 6

    

1 0 8 0 1 10 0 0 k + 6

− −

 

  a)

 

x + 3 y z  = 2x 2y + z  = 3x y + z  =

− −

−

11 2

−3 −

b)

3 2

x y

a)

 

 

x = y =

−8 −10

0 = k + 6

k + 6 = 0

3x + 2 y z  = 5 2x + y +  z  = 7 x 2y + 3 z  = 1

−

−

z 

kx  + y + z  = 1 x +  ky  + z  = 1 x + y  + kz  = 1

b

b)

 

• • • c + b = 87 c + a   = 123 a + b = 66

−

−6

x + 2y + 8 z  = 0 x 3y 7z  = 0 x y z  = 0

− − − −

x + y + kz  = 2 3x + 4 y + 2 z  = k 2x + 3 y z  = 1

c

 

  k

• • •

a

c)

k  =

    

0 1 1 87 1 0 1 123 1 1 0 66

 

1 0 0

−

a b c

       

 



1 1 0 0 1 1 0 0 1

 

87 123 66

 =

1 1 0 66 1 0 1 123 0 1 1 87



1 0 66 1 1 87 1 1 57

c  = 72kg b  = 15kg

    ·   

0 1 1 1 0 1 1 1 0

       

1 1 0 66 0 1 1 87 1 0 1 123



1 1 0 66 0 1 1 87 0 0 2 144



   

1 1 0 66 0 1 1 87 0 0 1 72

    ·    a b c

66 87 72

 =

a + b = 66 b + c = 87 c = 72

a  = 51kg

• • • • a

e

� � �

1 5

−

1 50 3 130

� � 

1 0

1 5

a + e = 50 5a 3e   = 130

−

1 3

− 1 8

�·� � � � a e

50 120

− −

 =

� � 

50 130

1 1 50 0 1 15

� � 

1 0 35 0 1 15

�

� �·� � � � 1 0 0 1

a e

�

 =

35 15

a = 35 e = 15

• • •

a

b

c

• • •

     

2a + b + c   = 150 4a + 3b   = 240 5b + 3c   = 350

c + b + 2 a   = 150 3b + 4 a   = 240 3c + 5b   = 350

1 1 2 0 3 4 3 5 0

    ·    c b a

 =

150 240 350

 

1 1 2 150 0 3 4 240 3 5 0 350



R$ 50

 

    

1 0 0 1 0 0

1 1 0 3 0 2 5 3 13

2 4 6

150 240 100

200 50 390

   

 

1 0 0 0 1 0 0 0 1

− −

− −



    

1 0 0 1 0 0

1 1 0 3 0 1 5 3 1

2 4 3

− −

200 50 30

− −

150 240 50

    

    ·    c b a

 =

    

1 1 0 1 0 3

1 0 0 50 0 1 0 40 0 0 1 30

2 3 4

150 50 240

− −

 

 

50 40 30

R$ 30 R$ 40