A) Solve the equations for x and y (x−12y) + (x−13y)i = 2+13i b) Find the eigenvectors and eigenvalues for the given matrix, A A = [[2, 1], [1, 2]]

The given equation is $(x - i2y) + (y - i3x) = 2 + i3$.
Combine the real terms and imaginary terms:

$$(x + y) + i(-2y - 3x) = 2 + i3$$

Equating the real parts:

$$x + y = 2 \quad (\text{Equation 1})$$

Equating the imaginary parts:

$$-3x - 2y = 3 \quad (\text{Equation 2})$$

From Equation 1, express $y$ in terms of $x$:

$$y = 2 - x$$

Substitute this expression for $y$ into Equation 2:

$$-3x - 2(2 - x) = 3$$ $$-3x - 4 + 2x = 3$$ $$-x - 4 = 3$$ $$-x = 3 + 4$$ $$-x = 7$$ $$x = -7$$

$$y = 2 - (-7)$$ $$y = 2 + 7$$ $$y = 9$$

The solution for $x$ and $y$ is:

$$\boxed{x = -7, y = 9}$$