1. Calcular los determinantes: a) ∣∣∣∣ 3 4−2 7 ∣∣∣∣ . b) ∣∣∣∣x+ 1 −x−x x+ 1 ∣∣∣∣ . c) ∣∣∣∣cosα − senαsenα cosα ∣∣∣∣ . d) ∣∣∣∣z −1z z ∣∣∣∣ (z = cos ...

Vamos calcular os determinantes: a) ∣∣∣∣ 3 4 -2 7 ∣∣∣∣ O determinante dessa matriz é dado por: (3 * 7) - (4 * -2) = 21 + 8 = 29. b) ∣∣∣∣ x+1 -x -x x+1 ∣∣∣∣ O determinante dessa matriz é dado por: (x+1)(x+1) - (-x)(-x) = (x+1)^2 - x^2 = x^2 + 2x + 1 - x^2 = 2x + 1. c) ∣∣∣∣ cosα -senα senα cosα ∣∣∣∣ O determinante dessa matriz é dado por: (cosα * cosα) - (-senα * senα) = cos^2α + sen^2α = 1. d) ∣∣∣∣ z -1 z z ∣∣∣∣ (z = cos 2π/3 + i sen 2π/3) Substituindo o valor de z na matriz, temos: ∣∣∣∣ cos 2π/3 + i sen 2π/3 -1 cos 2π/3 + i sen 2π/3 cos 2π/3 + i sen 2π/3 ∣∣∣∣ O determinante dessa matriz é dado por: (cos 2π/3 + i sen 2π/3)(cos 2π/3 + i sen 2π/3) - (-1)(cos 2π/3 + i sen 2π/3)(cos 2π/3 + i sen 2π/3) = (cos^2(2π/3) + 2i sen(2π/3) cos(2π/3) + i^2 sen^2(2π/3)) - (-cos(2π/3) - i sen(2π/3) cos(2π/3) - i^2 sen^2(2π/3)) = (cos^2(2π/3) + 2i sen(2π/3) cos(2π/3) - sen^2(2π/3)) - (-cos(2π/3) - i sen(2π/3) cos(2π/3) + sen^2(2π/3)) = cos^2(2π/3) + 2i sen(2π/3) cos(2π/3) - sen^2(2π/3) + cos(2π/3) + i sen(2π/3) cos(2π/3) - sen^2(2π/3) = cos^2(2π/3) + cos(2π/3) - sen^2(2π/3) - sen^2(2π/3) + 2i sen(2π/3) cos(2π/3) + i sen(2π/3) cos(2π/3) = 1 + cos(2π/3) + 3i sen(2π/3) cos(2π/3). Espero ter ajudado! Se tiver mais alguma dúvida, é só perguntar.