Finding eigenvector for a specific matrix

Question

Consider the following $3\times3$ real matrix: $$ A=\begin{pmatrix} 1-a_1-a_2 & b_1 & c_1 \\ a_1 & 1-b_1-b_2 & c_2\\ a_2 & b_2 & 1-c_1-c_2\\ \end{pmatrix} $$ Note that $$ I_3-A= \begin{pmatrix} a_1+a_2 & -b_1 & -c_1\\ -a_1 & b_1+b_2 & -c_2\\ -a_2 & -b_2 & c_1+c_2\\ \end{pmatrix} $$ so it is easy to see that $\lambda=1$ is an eigenvalue of $A$. What is its corresponding eigenvector? If we consider a similar problem but for the $2\times2$ matrix $$ A= \begin{pmatrix} 1-a & b \\ a & 1-b \\ \end{pmatrix}, $$ then it is easy to see that $\lambda=1$ is an eigenvalue and its corresponding eigenvector is $$ \vec{v}= \begin{pmatrix} b \\ a \\ \end{pmatrix} $$ Can this result be generalized to $3\times 3$?