Lista GA

Gabarito da Lista de Exerc´ ıcios 1: Matrizes e Sistemas Lineares
Universidade Federal de Santa Catarina
Departamento de Matem´tica a MTM5512 - Geometria Anal´ ıtica 1. x = 2, y = −1 e z =

√
3

4.

2. x = 2.
3. x = −3 e y = 5.
4. a = 0, b = 2, c =

1
10
ed= .
3
3

5. x = 2, y = 1 e z = −5.
6. x = 2, y = −2, z = 5 e w = −10.
7. (a) A + B =

7 −4 −1
;
4
3 −5

(b) B + C =

5 2 −1
;
1 8
7

(c) A − C =

2 −6
0
;
3 −5 −12

(d) C − A =

−2 6 0
;
−3 5 12

(e) 4A − 3B + 5C =

−7 78 99
;
21 4 3

(f) 2B − 3A − 6C =

4 −77 −90
;
−18 −13 −16

−26
84 102
;
12 −10
6


7
3


(h) (A − C)t + B t =  −13 −1 .
−9 −11
(g) X =

1




8. (a) AB = 



(b) BA =

−11 −1
11 −13
9 11 −23 −18
−17 13 −3 −61
59 33 −97 −8




;



−60 −42
;
−29
49

(c) (BA)C =

6 −450
−205
129

;

(d) B(AC) =

6 −450
−205
129

;




(e) B A = 

 t t

(f) C t At B t =


−11
9 −17
59

−1
11
13
33 
;
11 −23 −3 −97 

−13 −18 −61 −8
6 −205
;
−450
129

(g) C 2 =

−8 28
−21 13

(h) C 3 =

−100 108
;
−81 −19

−524
−105

−6

(j) EF =  3
−6

−6

(k) F E =  −2
−8

(i) C 4 =

;

140
;
−419

−2 16

0 12 ;
−4 −4

3 12

0 −6 ;
8 −4

−2 1
4

(l) EI = E =  1 0
3
−2 2 −1

−2 1
4

(m) IE = E =  1 0
3
−2 2 −1
(n) BAC − C 2 + 3(BA)t =



;


;
−166 −565
.
−310
263
2

9. (a) det(A) = 3;
(b) det(B) = −11;
(c) det(C) = −27;
(d) det(D) = −15;
(e) det(E) = −8;
(f) det(CD) = 405;
(g) det(DC) = 405;
(h) det(C t ) = −27;
(i) det(4B) = −176;
(j) det(−2B) = −44;
(k) det(−3C) = 729;
(l) det(C + D) = 32.
10. (a) det(F ) = 5 det(B) = −55;
(b) det(G) = 5 · (−2) det(B) = 110;
(c) det(H) = 23 det(C) = −216;
(d) det(I) = 0;
(e) det(J) = − det(E) = 8;
(f) det(K) = 0;
(g) det(L) = 0;
(h) det(M ) = det(C) = −27;
(i) det(N ) = (−1) · (−3) · 4 · 2 = 24;