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Aula: Propriedades dos Determinantes

Atividade • 23 Questões

Q1:

Dado que 𝑛 = | | | | 6 βˆ’ 8 9 1 5 βˆ’ 9 βˆ’ 1 1 βˆ’ 7 2 βˆ’ 4 | | | | e π‘š = | | | | 1 8 βˆ’ 2 4 2 7 9 0 βˆ’ 5 4 βˆ’ 6 6 βˆ’ 3 5 1 0 βˆ’ 2 0 | | | | , encontre a relação entre π‘š e 𝑛 sem desenvolver qualquer determinante.

  • A 9 0 𝑛 = π‘š
  • B 1 8 𝑛 = π‘š
  • C 𝑛 = π‘š
  • D 1 5 𝑛 = π‘š

Q2:

Selecione o determinante que Γ© igual a | | | | 2 3 2 4 9 4 2 3 2 2 5 2 8 2 1 1 4 | | | | .

  • A 1 4 | | | | 1 3 2 4 9 2 1 3 2 2 5 2 3 2 | | | |
  • B 2 | | | | 1 4 9 3 2 2 1 2 5 3 2 2 2 3 | | | |
  • C 2 | | | | 1 3 2 4 9 2 1 3 2 2 5 2 3 2 | | | |
  • D 1 4 | | | | 1 4 9 3 2 2 1 2 5 3 2 2 2 3 | | | |
  • E 7 | | | | 1 3 2 4 9 2 1 3 2 2 5 2 3 2 | | | |

Q3:

Encontre, na sua forma mais simples, uma expressΓ£o para o determinante | | | | 9 βˆ’ 2 π‘˜ βˆ’ 2 π‘š 7 𝑛 βˆ’ 2 π‘˜ βˆ’ 1 βˆ’ 2 π‘š 7 𝑛 βˆ’ 2 π‘˜ βˆ’ 2 π‘š 9 + 7 𝑛 | | | | .

  • A βˆ’ 9 ( 9 + 1 8 π‘š + 7 𝑛 βˆ’ 2 π‘˜ )
  • B βˆ’ 8 1 βˆ’ 1 8 π‘š βˆ’ 7 𝑛 + 1 8 π‘˜
  • C βˆ’ 9 ( 9 + 1 8 π‘š + 7 𝑛 + 2 π‘˜ )
  • D βˆ’ 9 ( 9 + 1 8 π‘š βˆ’ 7 𝑛 βˆ’ 2 π‘˜ )
  • E βˆ’ 9 ( 9 + 1 8 π‘š βˆ’ 7 𝑛 + 2 π‘˜ )

Q4:

Selecione um fator do determinante | | | | π‘₯ βˆ’ 4 βˆ’ 8 π‘₯ βˆ’ 8 π‘₯ + 1 π‘₯ + 1 5 4 8 π‘₯ + 8 | | | | .

  • A π‘₯
  • B π‘₯ + 6
  • C π‘₯ + 9
  • D π‘₯ + 8

Q5:

Considere a equação | | | | 6 βˆ’ 2 0 1 1 βˆ’ 1 4 βˆ’ 1 8 1 4 βˆ’ 1 6 1 2 | | | | = 2 9 8 .

Encontre, sem desenvolver, o valor do determinante | | | | 2 βˆ’ 1 6 1 1 4 βˆ’ 1 4 βˆ’ 1 8 1 1 6 βˆ’ 2 0 | | | | .

Q6:

Escreva o determinante | | | | βˆ’ 4 βˆ’ 5 7 8 βˆ’ 4 1 6 5 2 2 3 8 | | | | na forma triangular superior e determine o seu valor.

  • A | | | | βˆ’ 4 βˆ’ 5 7 0 βˆ’ 1 4 3 0 0 0 9 | | | | , 504
  • B | | | | βˆ’ 4 βˆ’ 5 7 0 1 4 βˆ’ 3 0 0 0 9 | | | | , βˆ’ 5 0 4
  • C | | | | βˆ’ 4 0 0 βˆ’ 5 βˆ’ 1 4 0 7 3 0 9 9 | | | | ; 5 544
  • D | | | | βˆ’ 4 0 0 βˆ’ 5 βˆ’ 1 4 0 7 3 0 βˆ’ 9 | | | | , βˆ’ 5 5 4 4

Q7:

Em qual dos seguintes pares de pontos, a reta representada pela equação | | | | 𝑦 π‘₯ βˆ’ 8 3 βˆ’ 9 βˆ’ 5 0 βˆ’ 1 βˆ’ 1 | | | | = 0 passa por eles?

  • A ( βˆ’ 4 , βˆ’ 3 ) , ( 0 , βˆ’ 6 )
  • B ( βˆ’ 4 , βˆ’ 3 ) , ( βˆ’ 6 , 0 )
  • C ( βˆ’ 3 , βˆ’ 4 ) , ( βˆ’ 6 , 0 )
  • D ( 0 , βˆ’ 6 ) , ( βˆ’ 3 , βˆ’ 4 )

Q8:

Utilize as propriedades dos determinantes para calcular | | | | 6 𝑦 + 4 𝑧 5 π‘₯ βˆ’ 4 𝑧 5 π‘₯ 4 𝑧 + 5 π‘₯ 6 𝑦 βˆ’ 4 𝑧 6 𝑦 5 π‘₯ + 6 𝑦 0 4 𝑧 | | | | .

Q9:

Considere | | π‘₯ 𝑦 𝑧 𝑀 | | = 6 .

Encontre o valor de | | | ( π‘₯ βˆ’ 1 0 𝑦 ) 𝑦 ( 𝑧 βˆ’ 1 0 𝑀 ) 𝑀 | | | .

Q10:

Encontre o valor de | | | | βˆ’ 5 2 βˆ’ 4 0 5 0 3 0 0 | | | | .

Q11:

Encontre o valor de | | | | 4 1 βˆ’ 8 βˆ’ 6 3 6 0 0 0 | | | | .

Q12:

Encontre, sem expandir, o valor do determinante | | | | 1 8 0 βˆ’ 𝑦 𝑦 βˆ’ 𝑧 𝑧 βˆ’ 1 8 0 𝑦 βˆ’ 𝑧 𝑧 βˆ’ 1 8 0 1 8 0 βˆ’ 𝑦 𝑧 βˆ’ 1 8 0 1 8 0 βˆ’ 𝑦 𝑦 βˆ’ 𝑧 | | | | .

Q13:

Considere a equação | | | | 1 0 0 5 2 π‘₯ + √ 2 0 2 π‘š 1 + 2 π‘₯ | | | | = 0 . s e n c o s

Determine todos os valores possΓ­veis de π‘₯ dado que 0 ≀ π‘₯ ≀ 3 6 0 ∘ ∘ .

  • A 3 1 5 ∘ , 2 2 5 ∘ , 9 0 ∘ , 2 7 0 ∘
  • B 3 1 5 ∘ , 4 5 ∘ , 9 0 ∘ , 2 7 0 ∘
  • C 3 1 5 ∘ , 4 5 ∘ , 1 8 0 ∘ , 3 6 0 ∘
  • D 3 1 5 ∘ , 2 2 5 ∘ , 1 8 0 ∘ , 3 6 0 ∘

Q14:

Considere a equação | | | | π‘Ž 𝑏 𝑐 𝑒 𝑓 𝑔 π‘₯ 𝑦 𝑧 | | | | = 2 9 .

Calcule π‘Ž | | | 𝑏 𝑐 𝑓 𝑔 | | | βˆ’ 𝑏 | | π‘Ž 𝑐 𝑒 𝑔 | | + 𝑐 | | | π‘Ž 𝑏 𝑒 𝑓 | | | .

Q15:

Qual dos seguintes Γ© igual ao determinante de | | | | βˆ’ 9 𝑏 9 𝑏 7 𝑐 2 𝑐 βˆ’ 7 𝑐 βˆ’ 6 𝑑 βˆ’ π‘Ž 6 π‘Ž βˆ’ 3 𝑏 | | | | ?

  • A | | | | βˆ’ 1 8 𝑏 9 𝑏 7 𝑐 9 𝑐 βˆ’ 7 𝑐 βˆ’ 6 𝑑 βˆ’ 7 π‘Ž 6 π‘Ž βˆ’ 3 𝑏 | | | |
  • B | | | | 9 𝑏 βˆ’ 9 𝑏 7 𝑐 βˆ’ 7 𝑐 2 𝑐 βˆ’ 6 𝑑 6 π‘Ž βˆ’ π‘Ž βˆ’ 3 𝑏 | | | |
  • C | | | | βˆ’ 9 𝑏 βˆ’ 1 8 𝑏 7 𝑐 2 𝑐 9 𝑐 βˆ’ 6 𝑑 βˆ’ π‘Ž βˆ’ 7 π‘Ž βˆ’ 3 𝑏 | | | |
  • D | | | | 9 𝑏 βˆ’ 1 8 𝑏 7 𝑐 βˆ’ 7 𝑐 9 𝑐 βˆ’ 6 𝑑 6 π‘Ž βˆ’ 7 π‘Ž βˆ’ 3 𝑏 | | | |

Q16:

Sem fazer cΓ‘lculos, determine o valor do determinante | | | | 8 βˆ’ 3 βˆ’ 2 7 1 βˆ’ 8 2 4 βˆ’ 9 βˆ’ 6 | | | | .

Q17:

Utilizando as propriedades dos determinantes, encontre o valor de | | | | 5 0 π‘₯ π‘₯ 5 0 π‘₯ π‘₯ 5 0 π‘₯ βˆ’ π‘₯ | | | | . s e c s e c t g s e c s e c t g

Q18:

Calcule | | 1 1 1 2 1 4 1 5 | | + | | 1 2 1 3 1 5 1 6 | | + | | 1 3 1 4 1 6 1 7 | | + β‹― + | | 2 5 2 6 2 8 2 9 | | .

Q19:

Considere a equação | | | | 5 π‘Ž βˆ’ π‘₯ 5 𝑏 5 𝑐 5 𝑐 5 𝑏 βˆ’ π‘₯ 5 π‘Ž 5 𝑏 5 π‘Ž 5 𝑐 βˆ’ π‘₯ | | | | = 0 . Dado que π‘Ž + 𝑏 + 𝑐 = βˆ’ 1 , encontre seu conjunto solução.

  • A  βˆ’ 5 , 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) , βˆ’ 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) 
  • B  βˆ’ 5 , 2 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) , βˆ’ 2 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) 
  • C  βˆ’ 1 , 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) , βˆ’ 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) 
  • D  βˆ’ 1 , √ ( π‘Ž βˆ’ 𝑐 ) ( 5 π‘Ž βˆ’ 5 𝑏 ) , βˆ’ √ ( π‘Ž βˆ’ 𝑐 ) ( 5 π‘Ž βˆ’ 5 𝑏 ) 
  • E  βˆ’ 5 , √ ( π‘Ž βˆ’ 𝑐 ) ( 5 π‘Ž βˆ’ 5 𝑏 ) , βˆ’ √ ( π‘Ž βˆ’ 𝑐 ) ( 5 π‘Ž βˆ’ 5 𝑏 ) 

Q20:

Utilizando as propriedades dos determinantes, calcule o valor de | | | | βˆ’ 1 3 3 4 βˆ’ 8 βˆ’ 7 0 3 βˆ’ 5 | | | | .

Q21:

Utilize as propriedades dos determinantes para calcular | | | | 1 1 1 4 7 βˆ’ 1 4 βˆ’ 5 7 βˆ’ 3 9 1 4 | | | | .

Q22:

Use as propriedades dos determinantes para encontrar o valor de π‘˜ que faz π‘₯ um fator do determinante | | | | π‘₯ + 3 βˆ’ 3 4 βˆ’ 3 π‘₯ βˆ’ 1 βˆ’ 4 3 βˆ’ 5 π‘₯ + π‘˜ | | | | .

  • A4
  • B βˆ’ 4
  • C 4 3
  • D36

Q23:

Calcule | | βˆ’ 6 1 1 1 | | + | | βˆ’ 5 1 1 1 | | + | | βˆ’ 4 1 1 1 | | + β‹― + | | 1 0 1 1 1 | | .

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