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Álgebra Linear e Geometria Analítica - Anton - Álgebra Linear- Ed: 10º - Capítulo 2.2 - Ex. 4

Nos Exercícios 1-4, verifique que det (A) = d e t ( A^T ) .

4.

A = [[[,,,4,,,,2,,,-,1],[,,,0,,,,2,,,-,3],[-,1,,,,1,,,,,,5]]]

Passo 1

Para verificar se det (A) = d e t ( A^T ) . Vamos calcular o determinantes das duas matrizes.

Calculando o determinante da matriz A , pelo método de expansão de cofatores ao longo da segunda linha.

|[[,,,4,,,,2,,,-,1],[,,,0,,,,2,,,-,3],[-,1,,,,1,,,,,,5]]| = 0 xx C_21 + a_22 C_22 + a_23 C_23 = a_22 C_22 + a_23 C_23

C_22 = (- 1)^(2 + 2) M_22 = 1 * |[[,,,4,-,1],[-,1,,,,5]]| = (4 xx 5) - (- 1 xx - 1) = 20 - 1 = 19

C_23 = (- 1)^(2 + 3) M_23 = - 1 * |[[,,,4,2],[-,1,1]]| = - [(4 xx 1) - (2 xx - 1)] = - (4 - (- 2)) = - 6

det (A) = a_22 xx 19 + a_23 xx ( - 6 )

det (A) = 2 xx 19 + (- 3) xx (- 6) = 38 + 18 = 56

Passo 2

Calculado o determinante A^T , pelo método de expansão de cofatores ao longo da primeira linha.

|[[,,,4,,,,,,0,-,1],[,,,2,,,,,,2,,,,1],[-,1,,,-,3,,,,5]]| = a_11 C_11 + 0 xx C_12 + a_13 C_13 = a_11 C_11 + a_13 C_13

C_11 = (- 1)^(1 + 1) M_11 = 1 * |[[,,,2,1],[-,3,5]]| = (2 xx 5) - (1 xx - 3) = 10 - ( - 3 ) = 13

C_13 = (- 1)^(1 + 3) M_13 = 1 * |[[,,,2,,,,2],[-,1,-,3]]| = (2 xx - 3) - (2 xx - 1) = - 6 - (- 2) = - 4

det (A^T) = a_11 xx 13 + a_13 xx ( - 4 )

det (A^T) = 4 xx 13 + (- 1) xx (- 4) = 52 + 4 = 56

Logo os dois determinantes são iguais

Resposta

det (A) = det (A^T) = 56

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