Teste 4 2012.1

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UFRJ
Instituto de Matemática
Disciplina: Álgebra Linear II
Professor: Bruno, Luiz Carlos, Mário, Monique e
Paulo
Data: 24 de maio de 2012
Quarto teste
1. O determinante da matriz A =

3 3 2 −1
2 0 1 −2
3 3 3 −1
1 1 1 −1

é:
(a) 4
(b) 6
(c) 2
(d) 0
2. Seja β = {(1, 1, 1), (2,−3, 1), (1,−1, 1)} e A a matriz
mudança de base da base canônica para a base β. A
soma dos elementos da segunda coluna de A é:
(a) 0
(b) 1
(c) −1
(d) 2
3. Seja R a reflexão de vetores pela reta t(1, 2), t ∈ R. A
soma dos elementos da segunda coluna da matriz de
R na base canônica é:
(a) 75
(b) 15
(c) 125
(d) 35
4. Sabemos que T : R3 → R3 é linear, que β =
{~v1, ~v2, ~v3} é uma base para o R3, que ~v1 é autovetor
de T associado ao autovalor 3 e ~v3 é autovetor asso-
ciado ao autovalor −2. Qual matriz a seguir poderia
representar T na base β?
(a)
3 3 00 1 0
0 −2 −2

(b)
−2 −2 00 0 0
0 3 3

(c)
−2 −2 00 0 0
0 −2 3

(d)
3 1 10 λ 1
0 0 −2

5. Assinale a afirmativa falsa.
(a) Se A é diagonalizável (isto é, A =
MDM−1), então A é invertível (e A−1 =
MD−1M−1).
(b) Se An×n tem n autovalores distintos, então A é
diagonalizável.
(c) É possível construir uma matriz 3× 3 diagonali-
zável que tenha apenas 2 autovalores distintos.
(d) An×n é diagonalizável se e somente se a soma
das dimensões dos autoespaços de A for igual a
n.
6. Seja A =
 4 −3
6 −5
. A soma dos elementos da pri-
meira coluna de A10 é:
(a) −3068
(b) 3070
(c) 1024
(d) −1024
Gabarito Pág. 1
Gabarito dos 322 Testes Gerados
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Gabarito Pág. 4
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