Cho hệ phương trình tuyến tính AX = B (1) với \({A_{mxn}}(m > n),\overline A = (A\left| B \right.)\). Ta có:

Ta có:
\(\frac{{ - 1 + i\sqrt 3 }}{{1 + i}} = \frac{{\left( { - 1 + i\sqrt 3 } \right)\left( {1 - i} \right)}}{{\left( {1 + i} \right)\left( {1 - i} \right)}} = \frac{{ - 1 + i + i\sqrt 3 + \sqrt 3 }}{2} = \frac{{\left( {\sqrt 3 - 1} \right) + i\left( {\sqrt 3 + 1} \right)}}{2}\)
Đặt \(z = r\left( {cos\varphi + isin\varphi } \right)\), khi đó:
\(r = \sqrt {{{\left( {\frac{{\sqrt 3 - 1}}{2}} \right)}^2} + {{\left( {\frac{{\sqrt 3 + 1}}{2}} \right)}^2}} = \sqrt {\frac{{3 - 2\sqrt 3 + 1 + 3 + 2\sqrt 3 + 1}}{4}} = \sqrt {\frac{8}{4}} = \sqrt 2 \)
\(cos\varphi = \frac{{\sqrt 3 - 1}}{{2\sqrt 2 }};sin\varphi = \frac{{\sqrt 3 + 1}}{{2\sqrt 2 }}\) suy ra \(\varphi = \frac{{5\pi }}{{12}}\)
Do đó, \(z = {(\frac{{ - 1 + i\sqrt 3 }}{{1 + i}})^n} = {(\sqrt 2 )^n}(cos(\frac{{5n\pi }}{{12}}) + isin(\frac{{5n\pi }}{{12}}))\)
z là số thực khi và chỉ khi \(sin(\frac{{5n\pi }}{{12}}) = 0\) suy ra \(\frac{{5n\pi }}{{12}} = k\pi \) hay \(n = \frac{{12k}}{5}\) với k là số nguyên.
Để n nguyên dương nhỏ nhất thì k=5, suy ra n=12.

Ta có:
\(|w| = \left| {\frac{{z.{i^{2006}}}}{{\overline z }}} \right| = \frac{{\left| z \right|.\left| {{i^{2006}}} \right|}}{{\left| {\overline z } \right|}}\)
Mà \(\left| z \right| = \left| {\overline z } \right| = 5\) và \(\left| {{i^{2006}}} \right| = \left| {{i^{4.501 + 2}}} \right| = \left| {{i^2}} \right| = 1\)
Vậy \(|w| = \frac{{5.1}}{5} = 1\)

The given equation |z + 2 - i| + |z - 3 + 2i| = 1 represents the sum of distances from a point z to the points (-2, 1) and (3, -2) being equal to 1. The distance between the points (-2, 1) and (3, -2) is sqrt((3 - (-2))^2 + (-2 - 1)^2) = sqrt(25 + 9) = sqrt(34). Since sqrt(34) > 1, there is no solution for z that satisfies the equation. Therefore, the set of complex numbers z that satisfy the equation is empty. However, given the answer options, "All of the other answers are incorrect" is chosen, assuming the question is flawed.