Verifique que det (A B) = d e t ( B A ) e determine se vale a igualdade det (A + B) = det (A) + d e t ( B ) .

[[A,=,([[-,1,8,2],[1,0,-,1],[-,2,2,2]]),B,=,([[2,-,1,-,4],[1,1,3],[0,3,-,1]])]]

Passo 1

Dadas as matrizes A = ([[-,1,8,2],[1,0,-,1],[-,2,2,2]]) e B = ([[2,-,1,-,4],[1,1,3],[0,3,-,1]]) vamos primeiro determinar os produtos A B e B A .

  • Resolvendo A B
  • A B = ([[-,1,8,2],[1,0,-,1],[-,2,2,2]]) ([[2,-,1,-,4],[1,1,3],[0,3,-,1]])

    A B = ([[-,2,+,8,+,0,-,2,+,1,+,0,6,+,2,+,0],[1,+,0,+,0,-,3,+,4,+,0,9,+,8,+,0],[-,4,+,2,+,0,0,+,0,+,0,0,+,0,+,2]])

    A B = ([[6,15,26],[2,-,4,-,3],[-,2,10,12]])

  • Resolvendo B A

B A = ([[2,-,1,-,4],[1,1,3],[0,3,-,1]]) ([[-,1,8,2],[1,0,-,1],[-,2,2,2]])

B A = ([[-,2,-,1,+,8,16,-,0,-,8,4,+,1,-,8],[-,1,+,1,-,6,8,+,0,+,6,2,-,1,+,6],[0,+,3,+,2,0,+,0,-,2,0,-,3,+,4]])

B A = ([[5,8,-,3],[-,6,14,7],[5,-,2,-,5]])

Passo 2

Vamos usar o Teorema de Laplace e verificar se os determinantes são iguais:

det (A B) = 6 |[[-,4,-,3],[10,1,2]]| - 15 |[[2,-,3],[-,2,1,2]]| + 26 |[[2,4],[-,2,1,0]]|

det (A B) = 6 ((- 48) - (- 30)) - 15 ((24) - (6)) + 26 ( (2 0) - (- 8) )

det (A B) = - 108 - 270 + 728

det (A B) = - 66

det (B A) = 5 |[[1,4,7],[-,2,-,5]]| - 8 |[[-,6,7],[5,-,5]]| - 3 |[[-,6,14],[5,-,2]]|

det (B A) = 5 ((- 70) - (- 14)) - 8 ((30) - (35)) - 3 ((12) - (60))

det (B A) = - 2 80 + 40 + 144

det (B A) = - 66

Passo 3

Vamos achar agora a matriz A + B

A + B = ([[-,1,8,2],[1,0,-,1],[-,2,2,2]]) + ([[2,-,1,-,4],[1,1,3],[0,3,-,1]])

A + B = ([[-,1,+,2,8,+,(,-,1,),2,+,(,-,4,)],[1,+,1,0,+,1,-,1,+,3],[-,2,+,0,2,+,3,2,+,(,-,1,)]])

A + B = ([[1,7,-,2],[2,1,2],[-,2,5,1]])

Passo 4

Vamos agora comparar d e t ( A + B ) e det (A) + det (B)

det (A + B) = 1 |[[1,2],[5,1]]| - 7 |[[2,2],[-,2,1]]| + ( - 2 ) |[[2,1],[-,2,5]]|

det (A + B) = ((1) - (1 0)) - 7 ((2) - (- 4)) - 2 ((10) - (- 2))

det (A + B) = - 9 - 42 - 24

det (A B) = - 75

det (A) = ( - 1 ) |[[0,-,1],[2,2]]| - 8 |[[1,-,1],[-,2,2]]| + 2 |[[1,0],[-,2,2]]|

det (A) = (- 1) ((0) - (- 2)) - 8 ((2) - (2)) + 2 ((2) - (0))

det (A) = - 2 + 4

det (A) = 2

det (B) = 2 |[[1,3],[3,-,1]]| - ( - 1 ) |[[1,3],[0,-,1]]| + ( - 4 ) |[[1,1],[0,3]]|

det (B) = 2 ((- 1) - (9)) + ((- 1) - (0)) - 4 ((3) - (0))

det (B) = - 20 - 1 - 1 2

det (B) = - 33

det (A) + det (B) = 2 - 33

Logo, det (A + B) != det (A) + det (B)

Resposta

Logo, det (A B) = d e t ( B A ) e det (A + B) != det (A) + det (B)

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