The real part of a complex number z = a + bi is a. Here, the real part is -3.

|z|² = (1)² + (1)² = 1 + 1 = 2.

The argument of a complex number z = a + bi is given by θ = tan⁻¹(b/a). Here, θ = tan⁻¹(1/1) = π/4.

z^3 = (1 + i)^3 = 1 + 3i + 3i^2 + i^3 = 1 + 3i - 3 - i = -2 + 2i.

The conjugate of a complex number z = a + bi is given by z* = a - bi. Thus, the conjugate of z = 1 + i√3 is 1 - i√3.

z^2 = (1 + i√3)^2 = 1^2 + 2(1)(i√3) + (i√3)^2 = 1 + 2i√3 - 3 = -2 + 2i√3.

The modulus of z = 1 + √3i is |z| = √(1² + (√3)²) = √(1 + 3) = √4 = 2. The square of the modulus is 2² = 4.

z² = (2 + 2i)² = 4 + 8i - 4 = 8i.

The argument of z is given by tan^(-1)(2/2) = tan^(-1)(1) = π/4.

z^3 = (2 + 2i)^3 = 8 + 12i - 12 - 8i = 8.

Using Euler's formula, z = 2(cos(π/3) + i sin(π/3)) = 2(1/2 + i√3/2) = 1 + √3i.

The modulus of z is given by |z| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.

The conjugate of z = 4 - 3i is z* = 4 + 3i.

z = 4(cos(π/3) + i sin(π/3)) = 4(1/2 + i√3/2) = 2 + 2√3i.

The argument of z = 5 + 12i is θ = arctan(12/5).

z = 5(cos(π/3) + i sin(π/3)) = 5(1/2 + i√3/2) = 2.5 + 5i√3/2.

z1 + z2 = (2 + 2i) + (1 - i) = 3 + i.

z1 * z2 = (2 + 2i)(2 - 2i) = 4 - 4i^2 = 4 + 4 = 8.

z1 + z2 = (2 + 2i) + (3 - i) = 5 + i.

The determinant of an upper triangular matrix is the product of its diagonal elements: 1 * 1 * 1 = 1.

Det(G) = 0 because the first column has a zero entry, leading to a linear dependence.

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