I'm taking linear algebra II this semester and the course assumes that students have already covered complex numbers. Unfortunately I take my first analysis course, in which complex numbers are covered, this semester as well. I have already covered the basics by myself, but I'm not really comfortable with complex numbers yet, so right now I'm struggling with calculating the eigenvalues of a complex matrix.
Given the following matrix $A \in M_n(\mathbb{C})$ calculate the eigenvalues of A:
$$A = \begin{pmatrix} -1-i & -2-2i & -1-i \\ -i & 0 & -1 \\ 1+i & 2+2i & 1+i \end{pmatrix}$$
$A-\lambda I$ is:
$$A' = \begin{pmatrix} -1-i-\lambda & -2-2i & -1-i \\ -i & 0 - \lambda & -1 \\ 1+i & 2+2i & 1+i - \lambda \end{pmatrix}$$
After applying some elementary row and column operations that don't change the value of the determinant, I obtained, unfortunately I wasn't able to reduce A to upper triangular form:
$$A'' = \begin{pmatrix} -\lambda & -2-2i & -1-i \\ 1-i+ \lambda & - \lambda & -1 \\ 0 & 0 & - \lambda \end{pmatrix}$$
I think the determinant of $A''$ would then be: $$det(A'')=(-\lambda)^3 - (-\lambda(1-i+\lambda)(-2-2i)$$
My problem now is that I don't know how to treat the $\lambda$ in the multiplication of the second term. For example what is $-\lambda(1-i+\lambda)$? Is $\lambda$ treated as $(\lambda,0)$ or $(\lambda,\lambda)$?